Advances in Applied Mechanics Volume 16
Editorial Board T. BROOKE BENJAMIN Y.
c. FUNG
PAULGERMAIN
L. HOWARTH WILLIAM PRAGER T. Y . Wu HANSZIEGLER
Contributors to Volume 16 G. K. BATCHELOR ROBIN D. HILL
H. KOLSKY H. K. MOFFATT D. H. PEREGRINE GEORGE I. N. ROZVANY CHIA-SHUN YIH
ADVANCES IN
APPLIED MECHANICS Edited by Chia-Shun Yih DEPARTMENT OF APPLIED MECHANICS AND ENGINEERING SCIENCE THE UNIVERSITY OF MICHIGAN ANN ARBOR, MICHIGAN
V O L U M E 16
1976
A C A D E M I C PRESS
New York San Francisco London
A Subsidiary of Harcourt Brace Jovanovich, Publishers
COPYRIGHT 0 1976, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.
ACADEMIC PRESS, INC.
111 Fifth Avenue, New York, New York 10003
United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road,
London N W l
LIBRARY OF CONGRESS CATALOG CARD NUMBER:48-8503 ISBN 0- 12-002016-5 PRINTED IN THE UNITED STATES OF AMERICA
Contents
vii viii
LISTOF CONTRIBUTORS PREFACE
1
G. I. Taylor as I Knew Him
G.K.Batchelor Interaction of Water Waves and Currents
D. H.Peregrine 10 17 70 76 103 106 111
I. Introduction 11. Large-Scale Currents 111. Small-Scale Currents IV. Currents Varying with Depth V. Turbulence VI. Ship Waves References
Generation of Magnetic Fields by Fluid Motion
H.K. Moflatt I. Introduction 11. 111. IV. V.
Magnetokinematic Preliminaries Convection, Distortion, and Diffusion of 5Lines Some Basic Results The Mean Electromotive Force Generated by a Random Velocity Field VI. Braginskii's Theory of Nearly +xisymmetric Fields VII. Analytical and Numerical Solutions of the Dynamo Equations VIII. Dynamic Effects and Self-Equilibration References V
120 125 130 135 139 154 163 168 176
vi
Contents
The Theory of Optimal Load Transmission by Flexure George I . N . Romany and Robin D . Hill I. 11. 111. IV. V. VI. VII. VIII.
Introduction Static-Kinematic Optimelity Criteria-Plastic Design General Optimality Criteria-Elastic Design Optimal Flexure Fields-Basic Geometrical Properties Optimal Flexure Fields-Clamped Boundaries Optimal Flexure Fields-Mixed Boundary Conditions Optimal Flexure Fields-Further Developments Optimal Flexure Fields of Constrained Geometry List of Symbols References
i84 187 20 I 206 226 2 46 292 299 303 304
The Role of Experiment in the Development of Solid Mechanics-Some Examples H . Kolsky I. Introduction 11. Viscoelastic Behavior 111. Experimental Determination of Viscoelastic Response IV. Tensile Shock Waves in Rubber V. Plastic Wave Propagation VI. Experimental Studies in Dynamic Fracture VII. Conclusion References
309 3 16 322 345 349 354 364 365
Instability of Surface and Internal Waves Chiu-Shun Y ih I. General Introduction 11. Instability of Surface Waves 111. Instability of Internal Waves References AUTHOR INDEX SUBJECT INDEX
369 37 1 393 419 42 1 42 7
List of Contributors
Numbers in parentheses indicate the pngs on which the authors' contributions begin.
G. K. BATCHELOR, Department of Applied Mathematics and Theoretical Physics, University of Cambtidge, Cambridge, England (1)
ROBIN D. HILL, Faculty of Engineering, Monash University, Clayton, Victoria, Australia (183) H. KOLSKY,Division of Applied Mathematics, Brown University, Providence, Rhode Island (309)
H. K. MOFFAIT, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, England (1 19) Department of Mathematics, University of Bristol, Bristol, D. H. PEREGRINE, England (9) GEORGEI. N. ROZVANY,Faculty of Engineering, Monash University, Clayton, Victoria, Australia (183) CHIA-SHUNYIH, Department of Applied Mechanics and Engineering Science, University of Michigan, Ann Arbor, Michigan (369)
vii
Preface
This volume is dedicated to the memory of Sir Geoffrey Ingram Taylor (March 7, 1886-June 27, 1975), a great scientist and a wonderful man who was held in esteem and admiration by all who knew his work, and in affection by all who had the good fortune to enjoy his friendship. Much has been written about him and his work since his death, and immediately following these few pages is an intimate sketch of G. I. (for it is thus that many of us refer to him) by Professor G. K. Batchelor, G. I.’s associate for 30 years, who is also writing a more complete biographical article of him for publication by the Royal Society of London. Here I shall only say a few things in remembrance of him. His work was always marked by an originality of thought and a freshness of approach that continue to delight his readers, and a characteristic welding of analysis to experiment that is rarely attempted, let alone attained, by others. Since his collected works fill four large volumes, it is natural to wonder which of his works gave him the most satisfaction. In 1967, when he came to Ann Arbor to receive an honorary degree, I asked him that question, and he answered by singling out his work on the stability of Couette flows. At this, Professor A. M. Kuethe, who was walking with us, expressed some surprise, for he had thought G. 1,’s statistical theory of turbulence occupied that position of honor. I suppose G. I.’s answer might well have been different if I had asked him which of his works he considered the most important. But I do not think that question would have been as congenial to him. He considered himself an amateur in science. It is therefore not surprising that the spirit of adventure is often evident in his work, for one who works out ofcuriosity and love of the subject is likely to explore where others dare not tread or, more often, do not think of treading. But if he called himself an amateur, the term was appropriate in the etymological sense only. The spirit of adventure was evident in his life as well. He learned to fly an airplane and to parachute when aviation was young, he rode viii
Preface
ix
balloons,t he sailed all over Europe, he skied when bindings were primitive and skis were without steel edges, and he tried (unsuccessfully) to waterski when he was eighty years old! On his holidays he was often accompanied by his wife (nee Grace Stephanie F. Ravenhill), who shared his interests in travels and was an ideal companion. In 1929 they visited Borneo and then Japan, where they slept in picturesque ryokans (wayside inns). In 1933 and 1934 they visited Canada. The trip in 1934 was one on horseback through the Canadian Rockies, taken with two friends and a cook. What he enjoyed most of his holidays was the beauty of nature and the feeling of solitude that it imparted. Once I showed him three volumes of old prints by Hiroshige, and the silent figures, the swaying willows, the blooming cherry trees, the temples, the bridges, and the waters gave him so much pleasure that he told Brooke Benjamin about them when he returned to England. The Japan G. I. knew was relatively unspoiled compared to the Japan of today, but the Japan Hiroshige depicted would have suited him even better. His love of beauty might well be inherited from his father, who was an artist. In 1967, while we were looking at a book of Camille Pissaro’s paintings, G. I. recalled a visit of the Pissaros (Camille and his son Lucien, who was also an artist) at his father’s home in St. John’s Wood in London. The Pissaros were in London in 1892 and 1897. Thus he was able to recall things that happened when he was six years or eleven years old. On Sunday afternoons at St. John’s Wood, G. 1,’s father would make pencil drawings of roses without using an eraser. G. 1. gave me one of these drawings, of two sprigs of roses. The roses look soft and moist, as if the dew had just evaporated from them. In G. I.’s living room was an oil painting by his father, a purplish landscape of Wales with wet sand on the beach: not so wet as to give perfect reflection,just wet enough to give the feeling of wetness. G. I. seemed to have inherited his father’s love of the beauty of Wales, for he had a cottage in Llanfair, Wales, where he kept his sailboat. G. 1.3excellent health and adventurous spirit were only part of his gift for happiness; his capacity for enjoying the simple things in life and a natural
t He once described the hazards in landing balloons. One pulled the cord (to let gas escape) in order to descend, discarded sand bags to let it rise again when it descended too fast or toward the wrong place, and repeated the procedure over and over until one succeeded in landing. In one such maneuver, after some sand was discarded, “the sand rapidly dispersed and fell more slowly than the balloon so, relative to us, it went upwards. Then as our downward velocity stopped and reversed the sand caught up and filled the air all round us. It was like a sand storm.”
X
Preface
detachment from the worldly aspirations that are a burden to less fortunate people were, I think, what gave his life the happiness that mere health and adventures could not have bestowed. His detachment arose not only from a natural simplicity but also from a desire to conserve time and energy to do what he wanted most to do. Indeed he received his numerous honors with a simple pleasure. But he never let these honors change him, and it is easy to believe that he could have been as happy without them. I shall now quote two letters from him, because these are interesting, and because they will make the points I have made, only more eloquently. The first was written on July 28, 1969, and the second on October 30, 1969. Many thanks for your letter and congratulations. I had just come back from Cape Breton Island when my 0. M.t was announced. I had a beautifully quiet time there at my cousin’s place which is on a dirt road 6 miles from the nearest paved highway: n o noise from cars or planes-only the wind in the trees and the surf on the shore. One could safely drink out of any of the streams and I used t o bathe in one close to where it ran into the sea. It has been so dry since I got back that I have been trying to keep the plants alive by watering them. I had a very pleasant time (contrary to my expectations) when I was received by the Queen and given the 0. M.insignia. I had expected officials all round, but I was introduced by an equerry who retired and shut the door, leaving me to talk with the Queen for a quarter of an hour. She is very easy to talk to and to listen to, and though nothing of any particular importance was said by either of us I thoroughly enjoyed the interview. I formed the impression that she is a very nice person whom one would much like to have as a friend under other circumstances. I also have been looking at problems of interest, in particular what is the mechanism which drives sap up a tree and sugar down from the leaves? It is a very interesting problem but I doubt whether it has much hydrodynamic content. I think that individualists like you and me should not feel the unimportance of what we d o compared with the work of the thousands in the space programmes. Of course our works are minute by comparison; but as long as we enjoy doing them that is the thing.
Your writing of Gaspe reminds me of a holiday Stephanie and I took in 1933, sleeping in a barn which was, I think, then the nearest building to the lighthouse at Gaspe Point. I tried bathing but it was terribly cold. . . .
Although G. I. was never in China he did have one connection with it. Mary Boole, his mother’s elder sister, married Howard Hinton. The Hintons’ grandson William$ and his wife were in China during the Sino-Japanese f “This isa very illustrious order, confined to 24 people who make outstanding contributions of a non-political kind, and it is certainly the highest distinction which has been conferred on him. His close friend Adrian (whom you may remember as a former master of Trinity) is also a member of the Order of Merit. He will be able to wear a very fine medal and ribbon around his neck on dress-up occasions in the future.” So wrote George Batchelor in a letter to me. William’s sister Joan has been in China for many years and, like him, is very sympathetic to the Chinese people.
Preface
xi
War, and stayed until many years after the Revolution of 1949. Indeed, at least one daughter was born to them in China. I once asked G. I. why the Hintons stayed so long in China, and he answered that they did so because they were idealistic. The mechanics community has lost a leading light and a living example of excellence, goodness, and the possibility of happiness. His life and work will forever be an inspiration, but those of us who knew him shall always miss him.
YIH CHIA-SHUN
Sir Geoffrey I. Taylor, 1886-1975
G. I. Taylor as I Knew Him
I first met G. I. Taylor in April 1945 when I arrived in Cambridge to work for a Ph.D. under his supervision. Our meeting place was his rather crowded room in the old Cavendish Laboratory in which one maneuvered around a wind tunnel. My immediate impression was of a kindly and unpretentious man with an informal manner who did not think it necessary to remove the bicycle clips from his trouser legs. He showed me the wind tunnel that D. C . MacPhail had designed and built in 1940 to have a low level of turbulence in the working section. No use had been made of the wind tunnel during the war years, and G. I. (as I later learned to call him) hoped that a new research student would be interested in an investigation ofgrid-generated turbulence to follow up some of the work he did in the late thirties. My own taste was for theoretical studies, but A. A. Townsend joined me as a research student working under G. I. in June 1945 and he began what proved to be a long and fruitful series of experiments on turbulence in this little wind tunnel with a working section about 160 cm long and 50 cm square in cross section. G. I. had no specific plans for our research and, while always interested in our results, was content to let us do what we wished. This suited both of us well, since we had done a little research before coming to Cambridge and were prepared to be independent. Some supervisors like their research students to be junior collaborators on some definite problem, but G. I. seldom needed or wanted a collaborator, junior or senior, and assumed that everyone was as happy to go his own way in research as he was. Three years later (in 1948) I did in fact collaborate with him on the calculation of the effect of wire gauze on small disturbances in the velocity of a uniform stream. We had each done a little previous work on the problem, and some new measurements of the effect of wire gauze on turbulence by Dryden and Schubauer stimulated us into looking for a complete mathematical solution. I saw then the difficulty ofcollaboration with G. I. His perception was incredibly quick, and I felt like a tortoise trying to play with a hare. 1
2
G. K . Batchelor
Thinking back to that first meeting with Taylor 31 years ago, it is difficult for me to appreciate that he was older then than I am now. He was presumably well past his research peak, but at the age of 59 he seemed tremendously fertile and inventive. He wrote and published over 60 papers in a period of 27 years after the end of the second world war, and this is the period in which I knew him and could observe him at work. In duration and achievement it is equivalent to a normal and successful scientific career. But by 1945 he had already published over 140 papers and had made outstanding contributions in meteorology, oceanography, theory of metal crystals, turbulent flow, gas dynamics, and a host ofpractical problems thrown up by the two wars. Whenever I am tempted to claim that I knew him well, I need to remember that his creative working life lasted from 1909 until he suffered a severe stroke in 1972 and that I saw nothing of his early and middle years in which the bulk of his work was done. G. I. was a member of a talented and interesting family. His mother’s father was George Boole, the mathematician who pioneered the study of symbolic logic and what we now call Booleian algebra. Boole was a selfeducated schoolmaster at Lincoln when he did his best-known work, and he was later appointed to the professorship of mathematics at Queen’s College, Cork. This great and good man, who was said to be “as innocent as a child,” apparently inspired feelings of love and reverence among his pupils, colleagues, and acquaintances. G. I. has written with affection about George Boole in an article prepared for the Dublin centenary celebrations of the publication of Boole’s “ Laws of Thought ” [published in Proceedings ofthe Royal Irish Academy (1954, pp. 6 6 7 3 ) and, in slightly different form, in Notes and Records ofthe Royal Society (Vol. 12, 1956, pp. 44-52)]. Boole married Mary Everest, a niece of Sir George Everest, who held the post of Surveyor-General of India and was one of the pioneers of geodesy; and they had five daughters: Mary, Margaret (G. I.’s mother), Alice, Lucy, and Ethel. George died in 1864, the year in which Ethel was born, and his wife and children moved to London where Mrs. Boole supported the family by working as a matron at a hospital. The third daughter, Alice, also possessed a mathematical mind and, despite having no more training than school Euclid, became interested in four-dimensional geometry and discovered for herself the six regular figures that can exist in four-dimensional space and their three-dimensional sections. Alice Boole dropped her mathematical hobby when she married, whereas George Boole was able to take up his professionally on his appointment to Cork; both were amateurs, in the sense of being independent and self-reliant and free to do what they wished, and G. I. liked, I think, to see himself as a scientific amateur in something of the same way. The fourth daughter, Lucy, became a professor of chemistryprobably the first woman to do so in England-at the Royal Free Hospital in London.
G . I . Taylor as I Knew Him
3
The youngest daughter, Ethel Lilian, had literary and musical ability. She married a Polish-born Russian revolutionary, W. Voynich, who later became a rare-book dealer in London and New York. Ethel Voynich’s first novel, “The Gadfly,” is a powerful and tragic story of young lovers involved in revolutionary struggles in Italy during the mid-nineteenth century, and, perhaps for reasons to do with the subject matter, is immensely popular in the socialist countries, especially in USSR where over 5 million copies have been sold and the author has been ranked with Dickens. Ethel Voynich moved to New York in 1920 and remained there until her death in 1960. There is a fascinating article by Anne Fremantle about her life and the remarkable success of “The Gadfly” in History Today (Sept. 1975). Margaret Boole married Edward Taylor, who was an artist, and the house in St. John’s Wood (in London) in which G. I. and his younger brother grew up was part studio, part home. G. I.%father designed the decoration of the public rooms of big ships and painted landscapes, but will perhaps be remembered most for his fine pencil drawings of flowers, many of which are on display in museums and galleries in Britain. G. I. may have acquired from his father his love of plants and gardens and an interest in rare specimens. In 1925 G. I. married Stephanie Ravenhill, who had been a school mistress; they had no children. The Taylors were keen sailing and traveling partners in the ten years after their marriage and made many enterprising journeys, including a voyage in their own sailing boat up the coast of Norway to the Lofoten Islands and overland expeditions to the interior of Borneo and some of the more remote parts of Japan. In the postwar period when I first knew her, Stephanie’s role was mistress of “Farmfield,” their home in Cambridge. She was very hospitable, and especially kind to newcomers like my wife and me. I recall their regular Christmas parties at Farmfield with hot punch and mince pies, very English in atmosphere and diverting for me as a raw and impressionable research student, since any one of the people with whom I was playing games might turn out to be a knight or an FRS or even both. I should say that I have learned much more about G. I.’s family in the course of tidying up the papers left after his death than from him directly. He was a shy and reserved man, and seldom spoke about such personal matters. In conversation he seemed to me to be not very articulate, although on paper he had a charming style and in his later years wrote a number of general articles about his life and work which make delightful reading. He enjoyed these reminiscences, especially those with a scientific content, and from them we get some insight into a happy, uncomplicated man with mainly conservative attitudes and a taste for thinking about concrete matters rather than abstractions. One thing is made abundantly clear from these articles written in later life, if it was not already evident from his scientific papers, and that is his love of
4
G. K . Batchelor
experiments, apparatus, and gadgets. He was, ofcourse, more than competent with mathematical manipulations and was supremely good at constructing theories, but it was a well-designed experiment that gave him the most pleasure. A visitor to his room in the Cavendish Laboratory would usually be shown the experiment in progress or some photographic results, and, although it was not always easy to understand what the idea was, the twinkling eyes and enthusiastic description made G. 1,’s feelings clear. G. I. did sometimes refer to himself as an applied mathematician (and he was, of course, using those words in the sense that is conventional in Britain), but his interest was very much more in the “application” than in the “mathematics,” as I think is evident from the following remark made in 1952 in what was probably the first of his reminiscing lectures (called “A Scientist Remembers”): “I think if I were to start again I should still try to be an applied mathematician, because the number of amusing activities to which mathematics can lead one is so great.” And elsewhere he says that he got his satisfaction chiefly “ from the interplay between applied mathematics and experiment.” Inventiveness in devising a simple experiment which reveals the physical process relevant to some phenomenon or which gives accurately a desired quantity was one of the hallmarks of his genius. I recall an example of this originality which is easily explained and which appears almost trivial-once you have “seen” it. In the early fifties G. I. was exploring various cases of diffusion and convection of salt dissolved in water in a tube and was led to think about the conditions for stability of the stationary liquid in a vertical circular tube when the salt concentration increases upward. It is not too difficult to solve the eigenvalue problem that gives the maximum concentration gradient compatible with stability, and, as was his custom, G. I. wished also to measure the critical value of the concentration gradient. There arises then the problem of setting up stratified fluid in a vertical tube and of varying the concentration gradient, presumably increasing it from zero until overturning first occurs. G. I. saw, and I do not think any period of thought was required, that the right way-the “obvious” way-to do the experiment is to connect the top of a long vertical tube of pure water to a reservoir of dyed salt solution of known concentration and then to measure the length of the column of dye when it has stopped being carried down the tube by overturning. I feel sure I could have thought of that myself; why didn’t I ? Most of G. I.’s experiments were done to test some idea or calculation after he felt he understood what was going on rather than to acquire data from which the understanding would come. This is probably a reflection of the quantitative and analytical character of most problems in mechanics. At all events, he usually knew in advance what to expect and was seeking a measurement for comparison with some theoretical prediction. Because his
G.I . Taylor as I Knew Him
5
insight was so acute, it seldom happened that the experiment failed to yield the expected result; and if there was some numerical disagreement, it could often be accounted for by some extraneous factor. The form of experimental check he liked best was the observation of a single number of which a theoretical value was already available, and I think he sometimes stopped making measurements as soon as he had obtained agreement with the theoretical value, with blithe unconcern for the rules about number of significant figures, theory of errors, etc. Sometimes his conscience seems to have pricked him, although not to the point of compelling stern action, for in several of his papers there is a gleeful admission that the agreement obtained is closer than is justified by the accuracy of the measurements or the theory (see, for example, the end of his second paper in 1950 on the blast wave from an intense explosion, where he estimates the initial velocity of rise of the luminous hemisphere left by the first atomic bomb explosion by regarding it as a vacuous bubble of radius equal to that where the density of the heated air is half that of the surrounding cold air-why a half?-and ends up with a value which differs by only 2% from the observed value, which itself is uncertain to within a much bigger range). As I have said, G. I. was seldom in doubt about the general validity of his theory, and I am sure that any apparently cavalier treatment of observations never led him to incorrect conclusions; but it gave his close associates some innocent fun. G. I.’s interest in science seems to have been aroused first by physical experiments. He records that he knew he wanted to be a scientist from the time when as a schoolboy he heard Oliver Lodge give the traditional Christmas lectures for children at the Royal Institution in London. The lectures and demonstrations were on wireless telegraphy, and they stimulated G. I. and his friends to construct a Wimshurst machine with which they generated weak X rays and photographed the bones in his mother’s hand. Boat building, sailing, and navigation were other early hobbies that developed his practical skills and the adventurous curiosity that I think was the driving force of his life. When he did these things he thought for himself, and the independence and originality which came to him as easily as breathing led to some improvement, some technical development, or some new concept. His love of small-boat sailing showed him that the conventional anchors were inconveniently heavy to handle, and in about 1933 he designed a new one which is a triumph of geometrical imagination and which has a holding power three times that of the best anchor of traditional type and the same weight. The merits of this so-called CQR anchor are now generally recognized, and it is widely used for small boats. An even more impressive example of his capacity for original thinking is his work as meteorologist on the “Scotia” expedition which was sent in 1913 to report on icebergs in the North Atlantic following the loss of the
6
G. K . Batchelor
“Titanic” due to collision with an iceberg. He was just 27; it was less than five years since the completion of his mathematics and physics course at Cambridge; and he had only recently taken up the study of meteorology. Yet this tyro foresaw that he would have an opportunity of measuring the vertical distributions of velocity, temperature, and humidity in the friction layer of the atmosphere and so of finding the rates of turbulent transfer. Singlehanded he organized and executed experiments involving the launching of kites and balloons carrying instruments from the deck of a sailing ship and flying them up to 6000 feet, and he produced a classic paper on turbulent transfer in the atmosphere which contained as incidentals the vorticitytransfer theory and an independent derivation of the “Ekman spiral distribution of mean velocity. It is easy to see how G. I. was drawn to science, but what made him choose the mechanics of fluids and solids for his life’s work? His third undergraduate year at Cambridge was spent in studying physics, and, being given a further scholarship by Trinity College, he stayed on at the Cavendish Laboratory in 1908. J. J. Thomson suggested he might like to test the recent suggestion that the energy of light waves is quantized by observing whether interference fringes continue to be formed by light when the intensity is exceedingly small and the incident quanta would be widely separated. G. I. did this, at his parents’ home in London with home-made equipment costing about two dollars, and described the result, namely, that the fringes were unaffected by the feebleness of the incident light, in his first paper, a onepage note published in 1909. That seems to have been the end of his association with “pure” physics, and his next research was the well-known investigation of the structure of shock waves, published in 1910. I believe a paper by Rayleigh drew his attention to this problem. The work evidently impressed people in Cambridge, for he was. awarded a Smith’s Prize for it and was elected to a Fellowship at Trinity College in the same year. In 191 1 a temporary readership in dynamical meteorology was set up at Cambridge with money provided by a wealthy professor at Manchester named Schuster who wanted to encourage a mathematician to study the subject, and G. I. was appointed, so far as I know without any previous experience or specialist knowledge (which, of course, would be inconceivable nowadays). This appears to have been the beginning of a major commitment to applied and classical physics which was maintained for the rest of his life. His experiences during the early part of the first world war, when he joined a group at Farnborough helping to put the new field of aeronautics on a scientific basis, may have consolidated his interest in mechanical subjects. In practical aerodynamics he certainly found plenty of scope for the theoretical analysis of quite new problems, some with the spice of adventure about them, as when he made what were probably the first measurements of the pressure distribution over a wing in full-scale flight. ”
G.I . Taylor as I Knew Him G. I. knew how to conserve his time and energy for the things that he was interested in. He tended not to get involved in affairs going on around him and stayed out of movements, administration, and all activities which men undertake collectively. He could not avoid being put on committees, but I think the only ones on which he played more than a listening part were technical committees such as those connected with the Aeronautical Research Council. He steered clear of teaching and was not tempted to write books or pedagogical articles; and when he did agree to write something with an educational purpose, such as his chapter on turbulence in “ Modern Developments in Fluid Dynamics,” it was usually done as a straightforward adaptation of his research papers. His research aids were simple and unpretentious, and were typified by those projection slides that he took to conferences; G. I. used to make these by writing or drawing with his fountain pen on one piece of glass 34 in. square and binding it with adhesive tape to a second similar blank piece of glass. Correspondence is an unavoidable part of research work, but G. I. kept the time spent on that to a minimum by carrying around letters in his pockets until they had either been lost or answered and by writing all replies briefly and by hand. He did not try to “ keep up with the literature” (and did not need to do so),and his knowledge of past work on a topic of interest to him was often patchy. I have the impression that, in the postwar years at least, he tended to avoid problems that other people were working on and to take up ones that were entirely new or had been mistakenly passed by. This was perhaps in keeping with his view of himself as an amateur in science. He had no wish to be a leader or to influence or enlighten people, and wanted only to continue to indulge his scientific curiosity in a way that would make minimum demands on other people. It is perhaps a significant comment on his way of life that, so far as I know, he never took leave of absence and never visited some other institution for the purpose of pursuing his research there (except when he went to Los Alamos for periods of several weeks for highly confidential work under wartime conditions). He would not have felt any need to get away from Cambridge, since he had no duties there; and it is unlikely that he felt any need for the stimulation of new associates and a new environment. He had ideas in plenty to investigate, and no need of facilities other than his oneroom laboratory and his one technical assistant, both located in the Cavendish Laboratory. He liked to travel, yes, but the notion of going away from one’s own institution to work would have seemed strange to him. If I had to sum up G. I.’s character and life in a few words I should describe him as a scientific “natural.” He had the gift of being able to see clearly how things worked-what in more impressive language we call physical insight-and he had a remarkable capacity for quantitative and analytical thought. Physical science was the obvious subject for his enquiring
8
G.K . Batchelor
mind, and he was drawn to the more applicable aspects linked with engineering, and to mechanics in particular, by the scope they offered for more adventurous activities. Meteorology and aeronautics were the first fields to attract him, and the state of their development 60 years ago made them just right for him, particularly aeronautics which was then in its pioneering phase. He was given the best possible scientific education and opportunities for research as a consequence of wise decisions first by Trinity College, which gave him an undergraduate scholarship and a research fellowship; second, by Cambridge University, which appointed him to the Schuster readership in meteorology at an early age; and third, by the Royal Society, which supported him as a Yarrow Research Professor from 1923 until his (official) retirement in 1952 and enabled him to devote all his time to research during those 29 years. All these factors and circumstances together provided the possibility of an unusually fruitful scientific career, and the additional ingredient of an uncomplicated, modest, contented, purposeful character, with supreme self-confidence which came early in life as a consequence of being able to exercise his innate skills, turned this possibility into a magnificent actuality. Opportunities were made available to him, and he took them all and used them to the fullest. I have never met a man who fitted so naturally into the pattern of his life. His character and his activities were perfectly matched, and this enabled him to live and work with freedom from the tensions, the maladjustments, and the pretentions that limit and handicap ordinary men. The result: a lovable man and a contribution to science of the highest quality which will be an inspiration for many generations. G . K. BATCHELOR
Interaction of Water Waves and Currents D. H. PEREGRINE Department of Mathematics University of Bristol. Bvistol. England
I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 A . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 B. SeaWaves . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 C. Coastal Waves . . . . . . . . . . . . . . . . . . . . . . . . . 12 D . Waves in Rivers and Channels . . . . . . . . . . . . . . . . . . 13 E . Hydraulic Breakwaters . . . . . . . . . . . . . . . . . . . . . 14 F . ShipWaves . . . . . . . . . . . . . . . . . . . . . . . . . . 15 G. Generation of Currents . . . . . . . . . . . . . . . . . . . . . 16 H . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 I1. Large-Scale Currents . . . . . . . . . . . . . . . . . . . . . . . . 17 A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 17 B. Waves on Uniform Currents . . . . . . . . . . . . . . . . . . . 18 C. Waves on Slowly Varying Currents . . . . . . . . . . . . . . . . 26 D . Steady Current, Varying with Distance along the Stream . . . . . . . 39 E . Steady Current, Varying across the Stream . . . . . . . . . . . . . 53 F. Flows with Significant Vertical Accelerations . . . . . . . . . . . . 63 I11 . Small-Scale Currents . . . . . . . . . . . . . . . . . . . . . . . . 70 IV . Currents Varying with Depth . . . . . . . . . . . . . . . . . . . . 76 A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 76 B. Infinitesimal Waves . . . . . . . . . . . . . . . . . . . . . . . 77 C . Finite-Amplitude Waves . . . . . . . . . . . . . . . . . . . . . 91 D . Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 E . Waves on Flow in Channels . . . . . . . . . . . . . . . . . . . 102 V. Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 VI . ShipWaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Notes Added in Proof . . . . . . . . . . . . . . . . . . . . 117
9
10
D . H.Peregrine
I. Introduction
A. SUMMARY The varied physical circumstances in which interactions between water waves and currents occur are described in this introduction. Different mathematical approaches, relevant observations, and experiments that are applicable to all or some of these physical Circumstances are described in the other sections. The paper has been written with gravity waves and currents such as those in seas or rivers in mind: thus there are only incidental references to the effects of surface tension and viscosity, which are of greatest significance for length scales of the order of centimeters and smaller. The emphasis is on waves and their interaction with preexisting currents rather than on wavegenerated currents, although these are mentioned where they are relevant. In all water wave problems approximations must be made to find mathematical solutions in order to gain physical understanding. Almost always the water is supposed to be inviscid and the flow irrotational. Here the first of these approximations is made but in few cases can the second approximation hold. Another common simplifying assumption is that the waves are of sufficiently small amplitude for the free surface boundary conditions to be linearized and evaluated at, or close to, the mean free surface. Most progress can be made in this subject with such a constraint, but wherever possible finite-amplitude effects are discussed. In order to get a reasonably wide class of solutions further approximations are necessary, the most important being for short waves and long waves, that is, for waves short (or long) compared with the length scale in which significant current variations occur. Sections I1 and 111 are on large- and small-scale currents, respectively. Much of the theory of water waves on large-scale currents only differs in detail from theory for any short waves in a moving medium (e.g., see Bretherton, 1971, for a review of linear theory). An adequate theoretical description was first given by Longuet-Higgins and Stewart (1960, 1961), who introduced the idea of radiation stress. Further improvement in our understanding has come from the use of Whitham’s method of averaging a Lagrangian and the concept of wave action. Since this is probably the most important field of wave-current interaction a number of simple situations are examined in detail in Section 11. Relatively little work has been done on small-scale currents, and Section I11 is mainly about waves in the presence of thin shear layers. Unlike some other common forms of wave motion, water waves involve water motion varying with direction perpendicular to the space in which
Interaction of Water Waves and Currents
11
they propagate. That is, water waves propagate on the surface of the water, but their motion also varies with the depth. Thus if there is current variation with depth it may affect the waves on the water surface. This is the topic of Section IV, which includes appreciable detail because of the possible applications to waves on streams and to the effects of a wind-driven surface current. In most applications the currents are turbulent and are approximated by a corresponding mean flow. However, there is interaction between waves and turbulence, so the few results are discussed in Section V. The paper concludes with another short section on the interaction of waves generated by a ship with the flow around it. A number of new results are incorporated in the text at various points. Particular examples are the errors involved in neglecting currents (Section II,B), the behavior of small-amplitude waves at a stopping point (Section I1,D) and of finite-amplitude waves approaching a caustic (Section II,E), and the surface layer solution for waves above a critical layer Section IV,B). B. SEA WAVES Over a large part of the world’s oceans and seas the spatial distribution of surface currents due to the tides and ocean circulation is on such a large scale that even the largest ocean waves are on an effectivelyuniform current, unless global propagation is being considered. It is mainly near continental margins and in shallow seas that currents influence waves significantly and this is discussed further in Section 1,C. However, the strong western boundary currents of oceans can and do strongly affect ocean waves. This is especially true of waves propagating onto and against such currents. They are shortened and steepened by the adverse current and refracted into caustics and foci by the shear at the currents’ boundaries. Examples of damage to ships by waves on the Agulhas current are mentioned in Section I1,E. Even on a uniform current difficult problems of practical importance arise. This is particularly so with nonlinear properties of waves such as the forces they exert on structures. With the increasing number of offshore structures the prediction of forces due to combinations of currents and waves is of growing importance. An appreciation of the problem may be gained from Hogben (1974). The surface drift caused by wind stress plays a role in wave dynamics as Banner and Phillips (1974) show (Section IV,C). It is probably also important in the generation of waves by wind; the tangential stress at the interface is influenced by its velocity and it in turn will influence the flow of air over
12
D. H. Peregrine
the wave. Phillips and Banner (1974) have studied this boundary layer in the water, but its implications for the wind-wave system need further study. Another effect, whose importance is debatable but not negligible, is the interaction between water waves. The interaction of short waves with much longer waves can be treated in the same manner as interaction with a current (Section II,F) and such effects are easily observed when a long swell meets short wind waves. Less commonly seen is the interaction of short waves with the velocity field of long internal waves (Section II,D) but Gargett and Hughes (1972) show photographs of wave patterns that are best explained in this manner. The shortest waves are capillary waves, and their ubiquity among wind waves is in part due to the steep gravity waves. The small radius of curvature at the crest of the steepest gravity waves and the surface tension there act rather like a moving pressure distribution and generate capillary waves. However, in Section II,F another mechanism for forming such waves is pointed out. Short gravity-capillary waves being overtaken by a larger gravity wave can be reflected near its crest and propagate away from it as capillary-gravity waves. The wavelength of this second class of waves is longer than those generated by the first mechanism.
C. COASTAL WAVES Water waves have their greatest economic importance when they arrive at coastlines. This is due to their ability to erode and build up land, their power to damage man-made structures, and the difficulties they cause in the handling of ships. Practically all the work to date on predicting and assessing coastal wave problems has neglected their interaction with currents, even though currents are often strong in coastal regions. There are two main reason for this neglect. One is that the transformation of waves due to lessening depth and their refraction and diffraction by underwater and coastal topography are often more important. The other reason is that the currents are often inadequately known. This is because of their complexity. Tidal currents vary in cycles so that in many parts of the world a good measurement of them requires recording for at least a lunar month. Furthermore, local winds, especially in storms, can produce currents comparable in magnitude with those due to tides in many places. Man-made structures are often associated with river mouths (which may be the reason for a harbor’s existence) and the river flow introduces further variability. To complicate matters still further, the mass transport associated with an irrotational wave train usually sets up mean currents in the region of the
Interaction of Water Waves and Currents
13
shore. For more detail, Longuet-Higgins (1972) gives an account of the longshore currents generated by waves and James (1974) gives further detailed calculations. The waves also generate rip currents and other return flows, which complete a cycle of interaction by influencing the incoming waves (Section 11,D). These effects are observed in hydraulic models, but sometimes efforts are made to minimize rather than measure them, especially if the wave-generated currents are unsteady. It is not clear whether attempts should be made to incorporate current effects into most coastal wave studies. The substantial variations in typical current fields mean that getting the basic data is very expensive unless there is a strong, welldefined current system that clearly cannot be ignored. On the other hand, it is advisable to estimate the “worst” conditions together with their probability of occurrence. As is indicated in Section II,F, the common impression that waves are higher at high tide than at low tide receives support from theory, but focusing effects are probably more important. For example, tide races, in which steep waves occur on strong currents around headlands and in channels, are due to waves, propagating partially against the current, being concentrated. Fortunately such concentrations of wave energy are usually in the strongest current, which is not usually a point where structures are erected. Indeed, wave action at the shoreline may be substantially overestimated in some circumstances if currents are ignored, since they can have a sheltering effect. Right on a beach, in the surf zone, there are appreciable problems in actually describing the waves. One promising model is to use the finiteamplitude shallow-water equations with bores fitted. In such a model a separation into waves and currents does not assist analysis. The relatively small scale of the currents also accentuates the difficulties. For example, rip currents might be expected to accentuate waves incident upon them; but observations indicate that waves on rip currents tend to be lower and break less. This is probably a case of diffraction being more important than refract ion. Two interesting minor points are a proposal by Dagan (1975) that short wave-long wave interaction may contribute to wave breaking (Section II,F), and an occasional wave feature in the backwash from surf that Peregrine (1974) interprets as being due to current shear in the vertical (Section IV,C). D. WAVES IN RIVERSAND CHANNELS Rivers normally have a nonuniform current distribution and this directly affects all the waves that occur on rivers (with the possible exception of flood and tide waves). If the currents are sufficiently swift then variations in the
14
D. H. Peregrine
river’s bed and banks (e.g., a projecting obstacle) may cause stationary waves on the surface, like ship waves. If such waves are caused by a constriction in the channel they are usually confined to the region of maximum current velocity and a train of several crests may be seen (Section 11,D). Stationary waves caused by a sluice or weir may have a very large amplitude (surface shear waves, Section IV,C). Shorter waves are commonly generated by the wind and by boats. The most usual interaction of these waves with a typical current profile (i.e., maximum velocity away from the shore) is for waves traveling upstream to be refracted toward the maximum current, while waves traveling downstream are refracted toward the bank (Section 11,E). Thus, in midstream the former waves may persist for a considerable time since they are not dissipated by interaction with the banks, while the latter waves soon meet the bank and decay rapidly. This also means that river banks may suffer more wave action from the upstream direction. The variation of current along a river also affects waves (Section 11,C). For example, a boat traveling upstream at a constant speed relative to the water produces waves of constant amplitude, but if they propagate upstream into a region of stronger current they may be considerably amplified. Thus, it is possible in such a region to see a boat pass traveling upstream and for the waves following it to increase continually in amplitude for a considerable time after the boat has passed. The author has experienced this while sculling and written a note about it (Peregrine, 1972). Similarly, the boat may generate waves that are “stopped (that is, propagating upstream but with a group velocity equal to the stream velocity). These waves, which appear to be moving since their phase velocity is upstream, persist until dissipated, which may take a surprisingly long time. ”
E. HYDRAULIC BREAKWATERS A hydraulic breakwater is simply an extensive current of water directed toward waves in order to stop them. It works; but for long waves very substantial currents are needed. The power needed to pump the water is such that it is rarely an economically feasible proposition. A considerable number of experiments have been performed-most designed to assess the power requirements of specfic designs. Evans (1955) gives an historical perspective as well as some experimental results. More recent reports of experiments are Nece ef al. (1968), Bulson (1963), and Williams and Wiegel (1962). A closely related device is the bubble or pneumatic, breakwater. The
Interaction of Water Waves and Currents
15
original idea was that a stream of air flowing up through the water would form a region with a lower effective density than water, thus reflecting some of the wave energy. In fact, the entrainment of water by the air results in an outward surface current from a line of bubble generators, which is effective in stopping waves. Evans (1955) also includes and compares results from a pneumatic breakwater. A more recent paper is that by Green (1961), which also has a summary of previous work. Naturally there is no need for a breakwater current to extend down to the full depth of the water if the incident waves are in “deep water.” Experiments such as Evans’ (1955) show that a surface current need only extend to a depth that is only a small fraction of a wavelength in order to stop waves, if it is strong enough. While there is linear and nonlinear theory available for waves on a slowly varying current that is uniform with depth (Section II,D), when the vertical structure of the current is also important results of linear theory are only sufficient to find the local wavelength, although this gives a first approximation to the stopping velocity (Taylor, 1955). Witham’s method of using an averaged Lagrangian may provide a way of finding the variation of wave amplitude in such cases.
F. SHIPWAVES The greatest interest in ship waves is in their contribution to resisting a ship’s motion. This may be assessed by measuring the waves radiated by a ship. However, some of the energy and momentum that may be assigned to the wave field close to the ship is lost from the wave field, for example, by wave breaking. On the other hand, the waves generated by a ship depend on the flow of water around it, and if this is altered (for example, by boundary layer suction) then so is the radiated wave pattern. The interactions between flow and waves are discussed in more detail in Section VI. Theoretical methods of estimating wave resistance are complementary to the measurements since these are usually made on ship models and need to be scaled for use in ship design. Most mathematical models assume inviscid irrotational flow, but it has been shown by experiment and theory that the wake and boundary layer lead to significant effects on the waves. These are not adequately described by increasing the size of the ship to account for the displacement thickness of the boundary layer. Many theoretical models involve several different approximations and considerable care needs to be taken to ensure consistency when proceeding beyond the first approximation. The further approximations usually involve wave-current interaction terms.
16
D.H.Peregrine G. GENERATION OF CURRENTS
The mass transport associated with water waves is of second order in the amplitude but still makes an appreciable contribution to currents in the vicinity of coasts. Details of the mass transport in a uniform wave train are calculated in Longuet-Higgins (1953) and some more recent work is in Sleath (1973, 1974). Mass transport is transformed into a current that is not directly coupled with waves whenever there is wave dissipation. Similarly, when waves gain or lose momentum because of interaction with currents there is a corresponding change in the current. This must be taken into account in any theory dealing with wave-current interactions in water of finite depth unless the waves have infinitesimal amplitude. Equations governing such an interaction are given in Section II,C, but there have been few direct applications to current generation, mostly to longshore currents. Another mechanism for generating a current from waves is given by Craik (1970). He describes a nonlinear interaction between two wave trains propagating over a depth-dependent shear flow. The idea is that the shear flow models the shear due to wind stress. The current generated has streamlines that superficially resemble those of vortices aligned with the wind. It is suggested that this mechanism may be partly responsible for Langmuir vortices, which are a similar feature observed in the sea. H. NOTATION
Notation is generally explained as it is introduced, except for some conventions that are uniform throughout the article. Coordinate axes are chosen with Oz vertically upward and Ox usually in the direction of the current if it is unidirectional. The plane z = 0 is a horizontal surface at or near the mean free surface. The current field in the absence of waves is U(r, t) with components ( U , V, W), and it is U + u with components (V + u, V + u, W + w ) in the presence of waves. Similarly the free surface is
although Z is zero sufficiently often that the symbol is used with other meanings. The wave frequency relative to a fixed reference frame is w and relative to the water is 0. The phase velocity relative to the water is denoted by c in Section 11, but elsewhere c is the phase velocity relative to the frame of reference. The wave number vector k is taken to be (k, 0,O) or (I, rn, 0)
Interaction of Water Waves and Currents
17
depending on circumstances, and 8 denotes the angle between k and U.The amplitude of the wave motion of the water surface is denoted by a. When tensor notation is used, Greek suffixes have the values 1, 2 and Roman suffixes 1, 2, 3, where ( X I , x2
9
x3)
= ( x , y, z ) .
The two-dimensional vector operator @/ax, a/ay, 0) is denoted by V,. In various places one symbol is used with different meanings to avoid using unusual letters or substantial numbers of suffixes. However, when results are cited from other works the notation is changed to agree with that in use in this paper.
11. Large-Scale Currents
A. INTRODUCTION In many instances of waves riding upon currents, the time and length scales determined by the current are many times larger than the period or wavelength of the waves. The natural assumption is to suppose that at any particular point the waves may have the same properties as a plane wave train on a uniform current, and further that the parameters describing the wave train, such as amplitude and wavelength, may vary slowly with the current. It is intuitively clear that such an approximation is likely to be effective, but the problem can be approached more formally, by requiring
where k and w are the wave number (= 2n/wavelength) and frequency (= 2n/period) of the waves. From ratios of such wave and current scales one or more small parameters may be constructed and formal expansions of variables in powers of a small parameter can be used to obtain solutions. It is expected, but not proven, that such solutions are asymptotic to exact solutions. This short-wave, or large-scale current, approximation has to be used in conjunction with solutions for uniform plane waves on water moving with uniform velocity. Such a flow is irrotational, and hence a velocity potential may be introduced to simplify the analysis, but it is still necessary to approximate to obtain water wave solutions. Some of these approximate solutions are briefly reviewed in Section II,B.
18
D. H.Peregrine
It is commonly the case in short-wave approximations that the solutions are singular on certain lines or points. In water wave examples it is not always clear which approximation is responsible for the singularity. It may be the water wave approximation, e.g., if a small-amplitude assumption is made, or it may be the short-wave approximation of a locally plane wave. For small-amplitude waves the plane-wave approximation can usually be improved in those cases where two possible solutions converge on the same singularity, one representing waves propagating toward it and the other representing waves propagating away from it. The resulting solution will describe waves being reflected. The maximum steepness of such a solution will then indicate whether the water wave approximation is sufficient or not. Specific examples are described in Sections II,D and II,E, but no such solutions have been produced for finite-amplitude waves, although a singularity, which requires such a description, is noted in Section II,E. The subject of this section is only one aspect of the problem of wave propagation in a slowly varying medium and most work on the subject is not specifically confined to water waves or to moving media. In particular, work published in the last decade on nonlinear short-wave problems, all originating from Whitham’s (1965a,b) method of averaging, has shed considerable light on the propagation of linear and nonlinear waves in nonuniform, slowly varying media. The concept of wave action, crystallized by Bretherton and Garrett (1968), is particularly valuable for moving media. An extensive and up-to-date account of the subject is Whitham (1974). Water waves are also influenced by the depth of water and where this too varies slowly its variation can usually be included. This is done in the rest of this section wherever it is convenient to do so. B. WAVESON UNIFORM CURRENTS In many applications of short-wave approximations, either the equations are linear, as in vacuum electromagnetic theory, or a linear approximation is an excellent first approximation, as in acoustics. In water wave problems a linear approximation can be quite sufficient, but it is more usually only a rough guide when waves have an appreciable amplitude and hence greater importance. Hence a few parameters, which are most often used for sinusoidal linear waves, are defined for nonlinear plane waves. If a wave is periodic in both space and time, then the physical variables describing it will all be functions of a phase
-
x = k r - ot + 6,
(2.2)
in which k, the wave number vector, is perpendicular to planes of constant
Interaction of Water Waves and Currents
19
phase (e.g., wave crests for water waves). Water waves are an example of modal waves, that is, waves that have structure in a dimension in which they do not propagate, in this case down into the water. Thus k is essentially parallel to the mean water surface and has no component perpendicular to it. The wave number k and frequency w are made unique by choosing x so that its period is 27r; they then correspond to the usual definitions of wave number and radian frequency for sinusoidal waves. The phase velocity c defined by
c=w/k, k = lkl, (2.3) is also defined for nonlinear waves. It is relevant to note that phase velocity is not a vector, e.g., the phase velocity along a line in the direction of a unit vector e is w/(k * e). If r is a position vector in a frame of reference in which water is moving with uniform velocity U, then the corresponding position vector r’ in a frame of reference moving with the water is given by r’ = r - Ut.
(2.4)
Thus a wave on moving water described by
f(k
- r - wt)
(2.5)
is also described by f(k * r’ + k * Ut - wt) =f(k
- r’ - at).
(2.6) Thus if any wave property (e.g., a dispersion relation) is given for still water for a wave of frequency a, the corresponding property for a wave on water in uniform motion is given by the relation 0
=w
- k * U.
(2.7)
A uniform plane wave train of infinitesimal amplitude, propagating over still water of uniform depth h, with vertical surface displacement
C = a exp[i(k * r’ - at + S)]
(2.8)
above the mean level, has velocity potential
’=
+
iaa cosh k(z h ) exp[i(k r’ - at k sinh kh
+ S)]
and dispersion relation rsz = g k tanh kh.
(2.10) In these expressions the physical quantity is the real part of a complex expression, and z is measured upward from the mean free surface. If surface
20
D. H.Peregrine
tension T is to be included, then g should be replaced by g + Tk2/p in the dispersion relation (2.10). The group velocity, the velocity of energy propagation, is
:$( 1 + sin2hkhZkh)
c =--
(2.11)
for gravity waves. This linear approximation is a good approximation when all three of the parameters ak, alh, and a/k2h3
(2.12)
are much less than one. For finite amplitude waves there are various different approximations, of which two are most relevant. A straightforward perturbation expansion in powers of ak gives the Stokes’ wave approximation, which is appropriate for water of moderate or great depth, specifically when a/k2h3is small. For shallow-water waves a more subtle expansion, balancing the effects of a/h against ak, is needed to produce the cnoidal wave solution (e.g., see Whitham, 1974). Such expansion procedures are cumbersome for dealing with the highest waves and usually separate approaches have been made to that problem. However, recently, computer-assisted calculations have enabled expansions to be carried out to high orders giving results for most of the range of possible periodic waves (Schwartz, 1974) and for the limiting case of the solitary wave (Longuet-Higgins and Fenton, 1974). For purely capillary waves Crapper’s (1957) exact solution covers the whole range of amplitudes. Stokes (1847) noted that there is ambiguity in defining “still water” for a finite-amplitude wave train on water of finite depth. The two natural definitions, (i) the average velocity is zero at any point that is always submerged, and (ii) the average flow of water through any vertical plane is zero, are not equivalent. This is clearly shown by considering the most general form of the velocity potential for a periodic wave. Since only the physical variables need be periodic (2.13) 4 = B * r - Yt + @(x), where y, is the phase (2.2). The constant fi corresponds to a uniform velocity.
The physical interpretation of y is less clear, but it contributes to the mean pressure and is thus related to the mean level of the water. If definition (i) for still water is chosen, then B = 0. On the other hand, with definition (ii), 2nio
Bh=
-j [j‘VO dr] dt. 0
-h
(2.14)
Interaction of Water Waves and Currents
21
This value of b is often called the mass transport velocity of the waves. The contribution to the integral comes mainly from the region above the lowest value of ( in the troughs of the waves. Thus for small amplitude waves it is of order a’ and can often be neglected. For deep water this ambiguity disappears since the still water at great depth provides a reference frame. However the mass transport is still nonzero. As an example of a finite-amplitude solution the first terms in a Stokes’ wave expansion are ak(3 - Ti) 4Ti cos 2% O(a’k’)],
+
C#J
=b
-r-
yt
+-aak
cosh k(z + h) sin sinh kh
I
+ h, sin 2% + 3ak 8‘Oshsinh2k(z 4kh
+
(2.15)
x
I.
O(a2k2)
(2.16)
and the dispersion relation
.
(a - b k)’ = gk tanh k(h
+ 9Ti) a’k’ + O(a4k4)], + b) 1 + (9 - 10Ti 8 T:
I
(2.17)
where To = tanh kh. The parameters k, w, a and fl, y, b define a specific wave train within a phase shift 6, and the dispersion relation (2.17) provides one equation between them. The choice of a frame of reference determines fl, e.g., definition (ii) for still water gives = tka’c coth kh
+ O(a3k2c).
(2.18)
Either y or b may be chosen arbitrarily but they must satisfy the relation y = $pz
+ gb + [(l - Ti)aZa2/4Ti]+ O(a3kaz)
(2.19)
obtained from the constant terms in Bernoulli’s equation. More details of finite-amplitude waves are given in Section II,C, which shows the advantage of leaving fl, y, b in these expressions. Now, consider infinitesimal waves on the surface of a uniform stream U . The dispersion relation (2.10) becomes (w - k U)’ = gk tanh kh
(2.20)
after using (2.7). This may be conveniently rewritten W =
+a(k)+k*U,
(2.21)
D . H . Peregrine
22 where
a(k) = +(gk tanh kh)’”.
(2.22)
A common and direct use of dispersion relations is to find the value of k once w is known (or vice versa) in order to calculate other wave properties. For example: (i) Measurements of [ ( t )at one point may be available and a measure of velocity fluctuations on the bottom may be required; or (ii) waves may be generated at a fixed frequency w, as in many experiments. In the absence of a current, k is determined uniquely but the direction of k is undetermined. In the k plane (i.e., a plane where k is a position vector) the locus of possible solutions is a circle. There is a greater lack of uniqueness in solving Eq. (2.21) for k when there is a current, even if U is known. The easiest way of appreciating the solution of the dispersion relation for k is to consider the intersection of the plane
m=w-k.U
(2.23)
m = ka(k)
(2.24)
with the surface of revolution in (k,m) space. [This is a development of the graphical method of solution given by Jonsson et al. (1970) for k parallel to U.] The general form of the locus of solutions for U # 0 is seen by noting that if k is perpendicular to U then the current does not affect the solution, while for k in any other direction, specified by a unit vector e, a diametral section of (2.24) yields a curve as shown in Fig. 1. The trace of a typical plane (2.23) is also shown, and four solution points A, B, C, and D in that diametral plane are labeled. The solution point A corresponds to waves with a component of k in the direction of the current, being swept along by it so that the measured frequency w is greater than the frequency a relative to the water. Similarly, B represents waves with a component of k opposed to the current direction traveling more slowly relative to a fixed observer so that w is less than a. These solutions effectively exhibit the Doppler effect, with appropriate corrections for dispersion. The solution represented by point C does not occur without the current, or for nondispersive waves. It corresponds to waves propagating against the current, in the sense that their crests move upstream, but their energy is being swept downstream. That is, - c < u c o s e < -c*,
(2.25)
Interaction of Water Waves and Currents
23
FIG. 1. Solution of the dispersion relation showing multiple values of k for given w, h, and U.
where 6 is the angle between k and U.These waves have to be generated on the current. The point D corresponds to waves with a
and
k-e<0.
(2.26)
They are waves with a phase velocity relative to the water in the - e direction on a current with a relatively strong component in the + e direction, that is, waves whose direction of propagation is partly upstream but which are being swept downstream faster than their phase velocity. For a sufficiently strong current, the solutions B and C may coalesce for some value of 6. It is easily shown that this occurs when cg + u cOS
e = 0.
(2.27)
In these circumstances the wave energy is either at rest or moving perpendicular to the current. This is just the property required of a hydraulic breakwater in order to stop incident waves being transmitted. It is convenient to call such a current velocity a “stopping velocity.” Since the phase velocity of gravity waves is always greater than their group velocity the waves’ crests would be progressing against the stream but the wave energy would not. This can give rise to a fascinating sight, either in a laboratory flume or on a river, of a small group of waves generated at an appropriate point, to all appearances continually moving but in fact just staying put. If cg +
u cos 6 < 0,
then solutions of the types B and C do not exist.
(2.28)
24
D. H . Peregrine
Another interesting case is when w = 0, that is, waves are stationary on the current although their energy is being swept downstream. The condition for stationary waves may also be written
+ u cos e = 0.
(2.29)
Such waves occur frequently, usually caused by a fixed obstacle or a boundary of the flow. The difference between the dispersion relation (2.20) for waves on a current and (2.10) for waves on still water can be an appreciable source of error if the presence of a current is overlooked. Such errors are greatest if relations between wave properties at the surface and on the bottom are used, for example, if measurements of [ ( t ) are used to deduce bottom velocities, or if measurements of pressure at the bottom are used to deduce surface wave amplitudes. [Jonsson et al. (1970) give one numerical example.] Taking the latter of these two examples, if a Fourier component of the pressure fluctuation at z = - h has amplitude p(w),then the corresponding surface amplitude component is ~ ( w=) p ( 0 ) cash khlpg.
(2.30)
The maximum errors will clearly occur when k is parallel or antiparallel to U. These are best displayed in dimensionless form. Introduce
R = w(h/g)'/2
and
F = U(gh)-'l2,
(2.31)
so that the dispersion relations (2.10) and (2.20) become
R2 = p tanh p (R - Fq)'
=q
tanh q,
(2.32) (2.33)
where kh is written p in the case of no current and q when a current is included. Thus if (2.32)is used in error for (2.33),the computed amplitude a, and the correct amplitude a2 are in the ratio u2/al = cash q/cosh p.
(2.34)
This has been computed and the relative error (a2 - al)/ai is shown in Fig. 2. For the case of small F and R > 1.5, it is sufficient to use the deepwater approximations for the dispersion relations, and the computation of this error is then a simple exercise. Inspection of Fig. 2 shows that adverse currents have the greatest effect. As an illustration, Table 1 displays the minimum period of waves for which a current of 0.5 m sec- can be ignored if errors are to be kept within 5% or within 20%. This velocity is typical of tidal currents in many parts of the
'
Interaction of Water Waves and Currents
- Oid
-0.3
-0.z
-41
0,’
0;1
25
0:3
0.1
-F
FIG.2. Relative error in surface wave amplitude calculated from bottom pressures due to ignoring a current component parallel with the wave direction.
world. Most of the combinations of depth and period are within the range that may be measured by this technique. For the shallowest depth given, much of the error comes from the size of the current velocity compared with the wave velocity, whereas at the other extreme the differing variation of pressure with depth is most significant. TABLE 1 MINIMUM PERIOD OF WAVES FOR WHICH A CURRENT OF 0.5 m sec- MAY BE IGNORED IN CALCULATING SURFACE AMPLITUDESFROM BOTTOM PRESSURE MEASUREMENTS IF ERRORS ARE TO BE LESSTHAN 5 AND 20% Depth (m)
Period with error of 5 % (sec) Period with error of 20% (sec)
1
2
5
10
loo
4.5 2.7
5.4 3.2
6.9 4.3
8.0 5.3
14 11
Simultaneous measurements of bottom pressure and surface elevation are reported by Draper (1957), and a comparison is made with the theoretical ratio, based on still water. Quite a wide scatter is shown. However, most of the results are within the range 1.00 < a2/ul < 1.16
(2.35)
and can be explained by currents setting against the waves with velocities up to 1.2 m sec- (i.e., assuming a depth of about 6 m, which is mentioned for
26
D. H . Peregrine
the two records shown). The measurements were made on Eastbourne pier. Charts of English Channel tidal streams (Hydrographic Department, 1973) indicate that currents in that region can reach 2.6 knots (1.3 m sec-l) for ordinary spring tides so that the scatter is adequately explained, except for the paucity of records with a z / a l < 1.00 corresponding to favorable currents. Maybe measurements were only made at certain states of the tide. For waves generated by wind, it is a common belief that if the wind is opposed to the current the waves are larger. Naturally this is most simply explained by the greater velocity of the wind relative to the water. However this is insufficient to explain experimental results such as those of Francis and Dudgeon (1967). Vincent (1975) points out that the effective fetch of the wind is increased when there is an opposing current, since the wave energy must travel correspondingly slower. This provides a satisfactory explanation for Francis and Dudgeon’s experiments. In particular, it shows that if U + cg = 0 then large wave amplitudes may be found at very short fetches. There are many other problems of waves on uniform currents, but most of them do not merit attention here since there is no important difference from the corresponding problem for still water. For example, the waves generated by an obstacle in a uniform flow are identical to those generated by the same obstacle moving at constant speed through still water. However, if there are also other waves on the flow that are incident on the obstacle, there may be significant interaction. This is particularly the case where the frequency of the waves is such that eddy shedding by the obstacle is enhanced. Recent experimental work on circular cylinders in a uniform flow by King et al. (1973) shows the wide range of frequencies at which vortex induced oscillations do occur, and work reported by Tanida et al. (1973) describes some of the effects of oscillating the cylinders. Experimental work to explore the effects of water waves is being initiated in view of its importance to large maritime structures. C. WAVES ON SLOWLY VARYINGCURRENTS Large-scale variation of a current can change all the parameters describing a wave train. Some aspects of these changes are relatively simple to interpret. For example, if the flow accelerates or decelerates, the frequency of the waves, w, will increase or decrease, the well-known Doppler effect. On the other hand, if waves propagate onto a faster or slower flow, the frequency will remain constant, but the wavelength will either increase or decrease. To a certain extent this is due to the extension or contraction of the water surface. If there is a nonzero angle between the wave number vector and the current then a change in current will lead to refraction of the waves.
Interaction of Water Waves and Currents
21
At first sight it might appear that the consequent amplitude changes might be found by assuming conservation of wave energy. This is not the case; there is a transfer of energy between the current and the waves. However, there is a conserved quantity. It is “wave action” and is equal to the wave energy density divided by the frequency of the waves relative to the water. In commencing a study of waves on slowly varying currents it is natural to start by specifying the current field: U = U(r, t).
(2.36)
However, in general this is not possible. As is described in Section II,B there is an ambiguity in the definition of still water for finite-amplitude waves, which is usually interpreted as a mass flow associated with the waves. Thus superposition of waves on a current changes the current and its determination becomes part of the problem. Whitham (1962) highlights this aspect of the subject and gives specific examples. There are two important situations where the mass transport has negligible effects on the current field. One is in deep water, and the other is for infinitesimal waves. For these the current field may be specified in advance. For a consistent approach the current field should satisfy the equations of motion, and this is essential for proving general mathematical results. Much published work assumes either that there is potential flow or that the flow satisfies the finite-amplitude shallow-water wave equations. However, in experimental or natural conditions the current field is usually turbulent and the mean flow U(r, t ) satisfies no simple set of equations. Whether or not it may be reasonable to neglect the effect of turbulence on the waves is a matter discussed further in Section V. It is likely to be impractical to include it in any detailed analysis, except in those cases where it is sufficiently strongly developed to have a marked dissipative effect. Although the flows considered in this section are assumed to satisfy the inviscid equations of motion, it is reasonable to apply the techniques described to any flow that is very nearly uniform over a distance of a few wavelengths, a time of a few periods, and to a depth of one wavelength, or to the bottom if that is less deep. To simplify presentation, the free surface of the flow without any waves is assumed to be approximately horizontal. Large-scale flows with appreciable surface slopes and vertical accelerations are considered in Section I1,F. The method that is most effective in describing wave propagation in slowly varying media is the use of an averaged Lagrangian developed by Whitham for nonlinear dispersive waves. [A full account is given in Whitham (1974).] Two other approaches are also valuable; one is direct integration with respect to z and averaging of the equations of motion; the other is to assume an appropriate form for the solution in terms of a pertur-
D. H. Peregrine
28
bation expansion and substitute in the equations of motion. All three have their value but the Lagrangian approach is most informative and is given correspondingly more attention.
1. The Averaged Lagrangian and Its Application A variational formulation of the water wave problem is provided by Luke (1967) for the exact irrotational case, and some indication of how to proceed for rotational flows. Appropriate approximate Lagrangians for long-wave equations are given by Whitham (1967a). An averaged Lagrangian is found by substituting an appropriate plane wave solution, or approximate solution, into the Lagrangian, integrating with respect to depth, and averaging over the phase X . The water wave Lagrangians all involve potentials so that appropriate parameters corresponding to /3, y, and b of the Stokes’ wave approximations (2.15) and (2.16) need to be included. Thus in general the averaged Lagrangian is a function Y(k, 0,a ; By Y, b ; U, h).
(2.37)
In particular examples it may be convenient to combine fl and b with U and h, respectively. The primary assumption is that if the wave properties are slowly varying-not only because of variations in U and h, but also due to slow variations in initial or boundary conditions-then the variational principle
(2.38)
6Y=O
applies for variations of all the wave parameters. Examples of averaged Lagrangians are given at the end of this subsection. The derivation of the Euler equations corresponding to the variational principle (2.38) is simplified by reintroducing the phase x, by the relations
k = VX
and
w = -ax/at,
(2.39)
and by introducing a pseudophase t,b defined by
p = vt,b
and = -at,b/at. (2.40) The second derivatives of these phases provide four consistency conditions
(aklat) + v W = 0, Vxk=O, (apiat)
+ vy = 0,
vxp=o,
(2.41) (2.42) (2.43) (2.44)
Interaction of Water Waves and Currents
29
which are sometimes called kinematic relations. There are then four Euler equations corresponding to variation with respect to a, 2, b, and $. The averaged Lagrangians do not include any derivatives of a and b, so that the corresponding Euler equations are simply (2.45) (2.46)
and are relations between the parameters of (2.37) involving no derivatives. Indeed, (2.45) corresponds to the dispersion relation and (2.46) to the Bernoulli relation (2.19) between p, y, and b. Since 2 and $ do not appear explicitly in 9, the Euler equations for their variations are at am
-, as ak
”(”)=V.
-, a9
”(”)=V.
at
ay
ap
(2.47) (2.48)
where d/ak and a/ap denote the gradient operator in k space and $ space, respectively. Equation (2.48) only occurs for finite-amplitudewaves in water of finite depth and is found to be an equation for the conservation of mass, including the effects due to changes of level and mass flow associated with the waves. By defining A = aZ/ao
(2.49)
to be “wave action density” and
B=-a~/ak
(2.50)
to be “wave action flux,” Eq. (2.47) takes the form (dA/at)+ V * B = 0,
(2.51)
of a conservation equation for wave action. Equation (2.47) is given in Whitham (1965b), but specific attention is drawn to the advantages of using wave action, when considering moving media, by Bretherton and Garrett (1968). A more general definition of wave action, but equivalent to (2.49) where waves are locally plane, given by Hayes (1970),allows the assertion of conservation of wave action in general conservative systems with periodic waves.
30
D . H . Peregrine
For linear waves the set of equations to be solved is (2.41), (2.42), the dispersion relation (2.45), and (2.51). The wave action density A = E/a
(2.52)
B = (U + c,)A = (U + c,)E/a,
(2.53)
and the wave action flux where E = ipga’ is the wave energy density, a the frequency of the waves relative to the water, and c, their group velocity relative to the water. The dispersion relation for linear waves has the form w =k
U + a(k, h),
(2.54)
so that Eq. (2.41) becomes
[i+
aa ah (U + c,) V k, = - -ahaxa - k
au,
(2.55)
after using (2.42). Another useful equation that may be obtained from the same three equations is [;+(U+c,).V
w=k.
]
-+-au aaah at
(2.56)
ah at
It is most easily obtained by first differentiating (2.54) with respect to time, and its real value lies in the fact that in many problems the right-hand side of the equation is zero. The solution of problems involving linear waves is eased by the absence of amplitude from Eqs. (2.41), (2.42), (2.54), (2.55), and (2.56). They are not all independent, as is clear from their derivation, but by choosing the most suitable of them it is often a straightforward matter to find w and k. Inspection of Eqs. (2.55) and (2.56) shows that their characteristics are given by the lines (drldt) = U + c,
.
(2.57)
If there are no simplifying features as in the examples of Sections II,D and E, numerical methods may be used. In general it is appropriate to calculate these characteristics or “rays,” using Eq. (2.55) for the components of k. Once rays are found, the wave amplitude is then derived from the wave action equation, which can be written
[i+
(U + CJ
);(I. *
+ [v . (U + C,)]f)
= 0,
(2.58)
which expresses the conservation of wave action between rays. The types of
Znteraction of Water Waves and Currents
31
solution that may be obtained in various circumstances are illustrated by examples in Sections II,D and E, where analytic solution is possible. One defect of this type of approximation is that sets of rays may meet and the solution is then singular with the amplitude of the waves becoming infinite. For example, this occurs at a caustic of rays where two sets of rays are tangential to a line in space, and also at a point where U + c, is zero and the rays are tangential to a line in space-time. Such singularities occur at points where the approximation of a locally plane wave is not valid. Typically there is reflection, and a different type of solution, valid in the neighborhood of the singularity, is required. Typical examples are given in following subsections. Even when such solutions are found, it is prudent to check that the parameters (2.12) are within reasonable bounds for infinitesimal theory. Singularities of this general nature also occur for finiteamplitude waves but the details are different, as is shown for one example in Section I1,E. So far, no corresponding solution valid near the singularity has been described. For finite-amplitude waves the dispersion relation includes the amplitude, so that it is not in general possible to solve for k and o independently. However, it is possible to use the dispersion relation (2.45), and the Bernoulli relation (2.46), to eliminate one variable each. [For example, Lighthill (1965) eliminates a in discussing two-dimensional deep-water waves and by retaining x as a variable finds a second-order equation for it.] For the resulting equations there are four sets of characteristics. Two have velocities close to f(gh)'I2 and correspond to propagation of changes in mean level and flow rate. The other pair of characteristics may be real, in which case their velocities are near U _+ c,, or they may be imaginary. (For infinitesimal waves these characteristics are real but coincident.) Equations with imaginary characteristics are of elliptic type and are effectively unStable for initial value problems. Lighthill (1965) shows that equations for deep-water gravity waves are elliptic and this ties in with other approaches to the stability of a plane wave train (Benjamin, 1967; Phillips, 1967). An unstable wave train does not necessarily break, but a certain class of modulations of the wave train can grow from an infinitesimal initial magnitude until they substantially modify the wave train. The stability of a wave train is unaltered by a current, and although most applications may be worked out in terms of an initially uniform wave train instabilities should not be ignored. In some examples, their rate of growth may be slow enough for them to be ignored; on the other hand, propagation times may be long enough for instabilities to significantly affect weakly nonlinear waves, for which one might at first sight expect linear theory to be adequate. Numerical integration is also likely to be difficult when the governing equations are unstable.
32
D. H. Peregrine
For capillary-gravity waves on deep water Lighthill (1965) using a thirdorder approximation shows that characteristics are imaginary except for the parameter range
- 1 + 2(3)- ‘ I 2 < T k 2 / p g < *.
(2.59)
For large-amplitude deep-water gravity waves an approximate Lagrangian (2.69) enables Lighthill (1967) to show that the characteristics become real for kH > 0.679,
(2.60)
where H is the crest to trough height. For gravity waves in a finite depth of water, Whitham (1967b) confirmed Benjamin’s stability analysis for two-dimensional disturbances by showing that characteristics are imaginary for kh < 1.36.
(2.61)
The analysis, using a third-order Stokes wave solution, is continued to threedimensional modulations by Hayes (1973), who indicates that waves are unstable in this approximation for all kh, but that for kh less than 0.5 the region of instability is small and a better approximation should indicate stability. Hayes (1973) also presents a different, and in some ways simpler, approach to manipulating the equations. By using wave action A as an independent variable, instead of wave amplitude, it becomes simple to eliminate o.By its definition A =aslam,
(2.62)
9 = A w - H(k, A),
(2.63)
which integrates to
where the “constant ” of integration is a Hamiltonian. Further, the dispersion relation is now a q a A = 0,
(2.64)
which is an explicit expression for w, =az/aA.
(2.65)
The k gradient of this expression for o, a Z z / a A ak, is a velocity, which Hayes calls the basic group velocity, and it is of value in
Interaction of Water Waves and Currents
33
interpreting the propagation of changes in wave action for a finite-amplitude wave train. It seems inappropriate to summarize this work in any more detail since applications to waves on currents have yet to appear; the interested reader should refer to the original papers and to Whitham (1974). However, it is appropriate to record the various forms of average Lagrangian that have been evaluated. For pure capillary waves on deep water, Crapper's (1957) solution gives
Y = 2T - pa2/k3 - T2k3/pa2.
(2.66)
A second-order solution for capillary-gravity waves on deep water gives 1 - 2u ~ + 3 5 1 -~ 1~ 1 0 ~+ 64 + 2 4 ~ '- 1 1( 261-~~+74u3 ~ 1) ~IC + 8 ) 3 ( 3 -
($-
1-
.)'+4
(2.67)
where K
= Tk2/pg,
(2.68)
and in both (2.66) and (2.67)the amplitude has been eliminated by use of the dispersion relation. These results are from Lighthill [1965; Eqs. (80) and (9111. For deep-water gravity waves of all amplitudes, Lighthill's (1967)approximation is
Y = (pg/8k2)(s2- s3 - s4),
(2.69)
s = a2/gk - 1,
(2.70)
with and the range of s from small-amplitude to maximum steepness waves is 0 I s I 0.2. To find the trough to crest height, Lighthill suggests the approximation s = k2a2,
(2.71)
where here 2a is that height. It would be worth reexamining this useful approximation in the light of the properties of high waves calculated by Schwarz (1974) and Longuet-Higgins (1975) (e.g., the quantity s has a maximum value of 0.193).
D. H. Peregrine
34
For gravity waves on finite depth of water, Whitham (1967b, 1974) finds, from the Stokes’ approximation,
(2.72) where the symbols are the same as in Eq. (2.15) and thereafter. There is no need for fi to be small in expression (2.72)and it could be used to represent the total current. In using most Lagrangians for waves on currents, o would be substituted by (o- U k). This method of using an averaged Lagrangian is justified by using a two-time (and length) scale expansion. Further approximations are possible. Chu and Mei (1970) discuss details for Stokes waves, and also (Chu and Mei, 1971) give specific examples, in particular showing how the instability of a uniform wave train develops into stronger modulations that settle down into groups of waves of permanent form. Whitham (1974, Ch. 14 and Sect. 15.5) shows how such an analysis can be derived from the Lagrangian approach. No applications have been given to waves on a variable current, but the solutions discussed in Sections II,D and E indicate that when singularities occur it may be because the basic assumption of a nearly plane wave is at fault, rather than too high a modulation rate.
-
2. Averaged Equations of Motion A more direct approach to waves on a current is to start with the equations of motion and to divide the velocity and other variables into mean and fluctuating parts. The equations may then be averaged over the phase of the fluctuating motion after integration with respect to z. Then each term needs to be identified with appropriate physical quantities such as the wave energy density E. The method is straightforward in concept but appreciable algebraic manipulation is needed. Details are not presented here, but are given in the book by Phillips (1966). [Mei (1973) gives a minor correction; a term p,, dd/dx, should be added to the left-hand side of his Eq. (3.6.11).] The advantage of this approach is that it is easier to appreciate the physical significance of terms and hence to make appropriate additions to the equations in order to account for wave dissipation, wave generation, or even wave breaking. The simplest case is for infinitesimal waves. The consistency conditions
Interaction of Water Waves and Currents
35
(2.41) and (2.42) and the dispersion relation are again applicable. Instead of the wave action equation an energy equation is derived: (2.73)
The term Sap is called the radiation stress (Longuet-Higgins and Stewart, 1960) and is given, to a first approximation, by (2.74)
where u, and p are the velocity and pressure fields associated with the waves. This expression indicates that Sapmay be considered as a wave momentumflow tensor, or alternatively as minus a wave Reynolds' stress tensor. Its product with the current gradient in (2.73) gives the rate of energy transfer between waves and current. The energy equation (2.73) is exactly equivalent to the wave-action conservation equation (2.58). Bretherton and Garrett (1968) show the equivalence for a current satisfying the finite-amplitude shallow-water equations. Most other relevant flows are such that the waves are effectively on deep water, and it is a simple corollary of their proof to show the equivalence for deep-water waves. For capillary-gravity waves, expression (2.74) for Sas is 1 + 3K
sinh 2kh
(2.75)
where K = Tk2/pg and La are the components of a unit vector in the k direction (Longuet-Higgins and Stewart, 1964). For gravity waves this may be written (2.76)
Radiation stress is a valuable concept when interpreting the generation of longshore currents by waves incident on a coast; see Longuet-Higgins (1972) for a detailed account. A somewhat different way of identifying terms in the averaged equations of motion is given by Hasselmann (1971).Emphasis is on the Eulerian view for the mass flow associated with the waves, that is, that it occurs only in the surface layer above the wave troughs. Particular attention is given to short waves riding on much longer waves, which are discussed further in Section ILF.
D. H . Peregrine
36
3. Perturbation Equations and Asymptotic Expansions
In a sense this title covers nearly all the work discussed in Section 11. Most of the plane-wave solutions used result from a perturbation expansion in powers of the wave steepness or similar parameter. The equations for the behavior of the wave trains can also be derived as the first terms of an expansion in powers of k-', or some other parameter expressing the small size of the waves relative to the scale of the current. Indeed, this section could have been introduced with a formal expansion in terms of E and k - l . Valuable though such an approach may be in improving the theoretical foundations of the equations, it seems inappropriate in this particular field, where in most instances a first approximation in terms of k-' is all that has been found, and experiment and observation are still needed to assess the value of theoretical results. The purpose of this subsection is to indicate where a direct approach has been used or could be used with advantage. This is at present confined to infinitesimal waves, whose motion is sufficiently small that it is a perturbation of the current satisfying linear equations of motion. In the body of the water,
+ (U - V)u + (u at
au
-
*
1 V)U + - v p = 0,
(2.77)
v.u=o,
(2.78)
P
with boundary conditions
(arm)
.
(2.79) + (U v)r = w, P = pgr, at the mean free surface if the vertical acceleration of the current is negligible (if not, see Section 11,F). An alternative is available if the current velocity field is also very weak. Then one may assume that the two velocity fields are simply additive with any interaction appearing as a second-order perturbation. Radiation stress was first introduced in order to give a physical interpretation of the results of such a perturbation analysis (Longuet-Higgins and Stewart, 1960). The weak-current approach is also used by Taylor (1962) and can be of considerable value since results can often be found without any assumptions about the scale of the current. In principle it is possible to solve any problem, but in practice the solutions are often singular, demonstrating that the implicit assumption of a weak interaction is not always tenable. Ideally one would like to solve Eqs. (2.7'7) and (2.78), with their boundary conditions (2.79), exactly for particular flows. Then no approximation about the scale of the current field is required. The only such solutions the author is aware of are for some flows of the form U = ( V ( z ) ,0, 0) (Section IV,B)
Interaction of Water Waves and Currents
37
and for stationary waves on a uniform flow bounded by vortex sheets (Section 111). Once it is assumed that the current varies on a much larger scale than the waves, examination of the perturbation equations and boundary conditions (2.77)-(2.79) shows that the only term involving derivatives of U is the third one in (2.77). Thus, any first approximation does not include it, and U then appears in the equations in the same way as for a uniform velocity. This suggests the most usual way of finding solutions, which exactly parallels the results described in Section II,C,l. A solution for a uniform current is chosen, such as a plane wave, and the parameters describing that solution, such as a and k, are thought of as functions, varying slowly with U.Typically they are expanded in an asymptotic series of the form
a = a.
a1 ++ a2 + k kZ -
(2.80)
and after substitution in the perturbation problems, coefficients of each power of k are set equal to zero. The actual details can vary appreciably and multiple-length scales may be introduced. If a plane wave is chosen as the assumed form of solution (or “ansatz”), then the first approximation gives an “eikonal” equation for the rays and the second approximation a “ transport” equation for the amplitude, corresponding to the wave action equation:An advantage of this approach is that for a velocity field that can be expressed in simple analytic terms higher approximations in k - can be readily found. The major advantage of choosing an “ansatz” and substituting it in the perturbation equations is in dealing with cases where the wave field is not locally a plane wave. (For example, however many terms in k - are taken, it is not possible to describe any reflection if the initial assumption is that there is a single wave train.) In particular, this approach can be used to describe regions where rays form a caustic, and the plane-wave approximation gives a singular solution for the amplitude. At least two types of caustic may be identified. If the rays forming the caustic are straight in its neighborhood, then the caustic is similar to a caustic in a uniform medium and is not directly affected by the current field. At the other extreme, if the caustic is caused entirely by the curvature of rays, it is the refracting effect of the current that is the prime cause of its existence. In this case the wave properties at the caustic depend on the value of the current and on its gradient. (Note that where the ray theory gives a good approximation the wave properties depend only on the value of the current.) Two very different examples of this latter class of caustic are discussed in Sections II,D and E.
D. H. Peregrine
38
It is instructive to note the analogy with short-wavelengthsolutions of the linear second-order ordinary differential equation, (dZUldX2) + N(X)U = 0.
(2.81)
When N ( x ) b 0,sinusoidal solutions with slowly varying amplitude are found by the well-known WKB (or Liouville-Green) method. These correspond to the ray solutions in our problems. The exponential solutions for N ( x ) 0 can be found in a similar manner and correspond to regions where waves do not penetrate. These are joined near points N ( x ) = 0 by connecting solutions in terms of Airy functions (e.g., see Lanczos, 1961).Such points correspond to caustics and solutions in the neighborhood of a simple caustic usually involve Airy functions, though not necessarily in the relatively direct form that is found for equations of the form (2.81). Similarly, when the function N ( x )has two zeros within a reasonably short distance, parabolic cylinder functions are appropriate for approximating solutions. Two examples of the corresponding problem, when the ray solutions indicate two adjacent caustics, are mentioned in Section I1,E. Another wave pattern that needs special treatment is a focus of rays. There is no simple analogy with ordinary differential equations. Most foci form as cusps of caustics, and Pearcey (1946) gives an appropriate “cusp” function together with numerical values. Uniform asymptotic expansions for a cusped caustic are discussed by Ludwig (1966). If rays form a perfect focus (i.e., all crossing at the same point) a solution of the form
-=
C anr”Jn(cosnO)eiot
(2.82)
n
may be used, and the asymptotic form of the Bessel functions used to match with the ray approximation at a sufficientlylarge value of r. It is particularly easy to find the amplitude at the geometric focus, r = 0, with expression (2.82), but Pearcey’s cusp function shows that the maximum wave amplitude is not usually at that point. A large-scale current distribution can cause waves to form different types of caustic. It is possible that there might be different types of focus. Waves patterns where the ray approximation gives a singularity in the amplitude, such as caustics and foci, have often been interpreted as regions where water waves break. This interpretation is only justified if a local approximation gives a large amplitude, and more significantly a high wave steepness, in its vicinity. Following this general discussion, the next two subsections consider two very simple examples of unidirectional flow. A variety of solutions can be found analytically, which illustrate some of the effects that can happen in a more complex problem. There appear to be very few examples published
Interaction of Water Waves and Currents
39
where more realistic flows have been tackled. The change in period of ocean waves, measured in Cornwall, due to tidal currents encountered in their passage over the continental shelf is one example calculated by Barber (1949) and is in satisfactory agreement with measurements.
D. STEADYCURRENT, VARYINGWITH DISTANCE ALONG THE STREAM
A current of the form
u = [U(X),0901
(2.83)
can describe the flow in channels and rivers where the velocity varies in response to the depth h(x) and slope of the bottom. The flow from a hydraulic breakwater may also be described in this way, although in practice such a flow is best confined to a surface layer, and (2.83) is then only a reasonable approximation for very short waves. A closely related flow can occur with a propagating wave, when
u = [U(X - Ct), 0, 01.
(2.84)
If C is constant this flow is of the type (2.83) in a frame of reference moving with velocity C. Examples are the surface currents due to internal waves or to long shallow-water waves. The form (2.84) may describe tidal currents, but more usually the tide is made up of more than one propagating mode. However, in all these possible applications it is likely that the basic longwave motion is of sufficiently small amplitude that different modes may be superposed. The simplest example is when waves travel perpendicular to the current, that is, k*U=O. (2.85) There is no interaction. A simple, less trivial example is when waves travel parallel to the current, so that k * U = +kU. (2.86) For definiteness we take k in the positive x direction so that if waves propagate against a current U is negative. Initially, attention is restricted to infinitesimal waves. Since the current is steady, or considered in a reference frame in which that is so, Eq. (2.56) implies that on rays dxfdt = U
+ cS,
w = const.
(2.87) (2.88)
D. H.Peregrine
40
The waves must vary slowly for the theory to apply so that there is little loss of generality in considering a steady state situation in which w is constant everywhere; the dispersion relation az = (w - U k ) z = gk tanh kh
(2.89)
is then the only equation needed to determine k and hence a. The ambiguities in solution of Eq. (2.89) mentioned in Section II,B are avoided by requiring continuity of solution as U and h vary. For numerical solutions this suggests that the numerical integration of a differentiated form of (2.89) is likely to be preferred to direct solution, though care should be taken near double roots. Explicit solutions can be found for deep-water waves, in which case it is usually more convenient to work with the phase velocity of the waves relative to the water, that is, c = a/k =
+ (g/k)’”
= g/o,
(2.90)
+ U ) = g(c + U ) / C 2
(2.91)
which with (2.89) gives w = k(c
as the equation for c. The conservation of wave action implies that the wave action flux B = pgaZ( u + c&2a
(2.92)
is constant and equal to its value at the point where the waves are generated. For deep-water waves this becomes a2 = 2B/pc( U
+ +c),
(2.93)
and there is no difficulty in finding the amplitude. Even where the depth is significant in the dispersion relation there is no difficulty in calculating the wave properties numerically. The results of such calculations are presented by Jonsson et al. (1970) in both tabular and graphical form for the case of steady channel flow, where U ( x ) h ( x )= Q.
(2.94)
Expression (2.92) indicates that the amplitude may become unrealistically large if either a becomes very large or U + cpapproaches zero. The former case does not arise for waves generated on still water, but can do so for waves initially on a current. Since large a implies very short waves the deep-water relations hold, and (2.90) shows that a large corresponds to c -+ 0. Typical curves representing Eq. (2.91) in the (U, c) plane clearly show that c -,0 only as U + 0 (Fig. 3). That is the situation in which -U>cc,,
(2.95)
Interaction of Water Waves and Currents
41
t1.0
FIG.3. Relation (2.91) between c and LI for deep-water waves for different values of w . Units are chosen with g = 1. The letters A, B, C, and D in the sectors shown refer to the four solutions in Fig. 1.
and the waves are being swept downstream although they propagate upstream relative to the water. As the water slows down the waves continually shorten and increase in steepness so that small-amplitude theory becomes invalid. In practice the waves break and little or no energy is propagated into still water. The other singularity when
u+c*=o
(2.96)
is more interesting. At such a point the wave number and frequency are both finite, and if a subscript 0 indicates still-water values and subscript 1 conditions at (2.96), then u1
= -2Cl 1 =
-3C 10,
k1 = 4 k 0 ,
61
= 260.
(2.97)
The velocity U1 is conveniently called the “stopping” velocity since for waves propagating against a stream it has the effect of stopping them. It is behind the idea of a hydraulic breakwater and can also be observed in rivers and in “tide races,” which occur off headlands or in channels where tidal currents are locally enhanced. Such a stopping velocity leads to very rough
D. H. Peregrine
42
water surfaces as the wave energy density increases substantially. Upstream of such points, especially if the current slackens, the surface of the water is especially smooth as all short waves are eliminated. As (2.97) shows, the current required is only one-quarter of the phase velocity for the waves in still water, because the waves have a smaller wavelength. Most accounts of this phenomena (e.g., Phillips, 1966) assume that, because of the singularity in amplitude, waves necessarily break before reaching the stopping velocity, finite amplitude effects only serving to give a different value to U1. This is not necessarily the case since the stopping velocity corresponds to a caustic of rays. There is a corresponding set of rays being swept downstream. It is thus possible for the waves to be reflected and swept downstream, as shorter waves, by the current. Details of this process are found by looking for a solution of the perturbation equations (2.77)(2.79) in the neighborhood of the stopping velocity. Specifically, by using a perturbation stream function of the form +(x, z)e- iOr,
(2.98)
the equations can be reduced to
xv:I)=o
in z < o ,
(2.99)
with the boundary condition
x 2a+/az = d d2$/az2
on z = 0,
(2.100)
where the operator
x
8 ax
= U(x)-
dU +- iw. dx
(2.101)
To solve these equations near the stopping point it is necessary to choose a simple form for U ( x ) that has a value and a gradient at that point that equal those for the actual velocity. Two possible choices are U(X) = - f q ( l
+ Bx)
(2.102)
and U ( X )= - f c l es”.
(2.103)
For both of these functions it is relatively straightforward to find the complex Fourier transform with respect to x of Eq. (2.99) and its boundary condition (2.100).If U were constant (= Ul), then the Fourier transform of Ic/ would be Y ( k , z ) = G(k,)ekz.
(2.104)
For small fix it is expected that the solution will be a perturbation of this
Interaction of Water Waves and Currents
43
solution, so it is assumed that the only relevant values of the transform variable k are those such that K
= (k - k l ) / k 1 6 1.
(2.105)
This assumption simplifies Eq. (2.99) to Laplace's equation and gives the form "(k, z ) = A(k)e".
(2.106)
With the further, short-wave assumption that
8 e ki,
(2.107)
the boundary conditions resulting from either (2.102) or (2.103) can be simplified to a first-order differential equation in k for kA(k). This can be integrated to give, near 8x = 0, $ = const x
j
OD
exp{k,(ix
+ z) + iP2kl K3/12 + ~ [ kB(ix , +
2)
- 8/41)
dK,
-m
(2.108) which may be rewritten in terms of the Airy function, I&y)
=A
exp[k,(ix + z ) ] Ai[(4kf8)1/3(x- iz + i/4k1)]. (2.109)
This is a local solution for the wave motion, valid near the stopping point. It has no singularity, indicating that reflection can occur. However, to be useful it needs to be matched to an outer solution corresponding to the two branches of Eq. (2.91) for c and the solution of (2.93) for the amplitude with B positive for waves traveling upstream, but with B of the same magnitude and negative for the reflected waves. Such matching can be done by using the asymptotic expansion of the Airy function. Since solution (2.109) is valid when px
e 1,
(2.110)
and the asymptotic expansion applies when
(4k18)1'3x% 1,
(2.111)
there is a matching region of overlap where 8-3
9 x3 % (4k3)-',
(2.112)
which is nonzero because of (2.107). The result, in terms of the amplitude a. on still water, is A = 47P(2kl /8)1/6ao ,
(2.113)
44
D. H.Peregrine
and the amplification in wave steepness is
0.536Ak1= 17.1(k,/~)1~6aoko,
(2.114)
where 0.536is the maximum value of the Airy function. This indicates that although the small-amplitude approximation may be uniformly valid, it is so, for waves starting on still water, only for extremely gentle initial waves, since even if the large parameter kl//3 is not very big the factor 17.1 is appreciable. Further light is shed on the matter by Fig. 4,which shows the variation of k, a, and ak with a logarithmic scale, according to the simple ray theory. The ak curve for the reflected waves is particularly interesting. It has a minimum value where U = 0.98U1 corresponding to an amplification of 33.6.This implies that any reflected waves continually increase in steepness as they are swept further away from the stopping point. Indeed, it is likely that the minimum will be obscured by the reflection process.
FIG.4. Variation of wave number, amplitude, and wave steepness for deep-water waves on a current, according to the ray theory, Eqs. (2.91) and (2.93). A subscript 0 refers to still-water conditions and subscript 1 to conditions at the stopping velocity, LT, = -&.
Interaction of Water Waves and Currents
45
Reflection has not been reported by experimenters with hydraulic breakwaters. The reason is easy to see. The reflected waves cannot propagate to still water and have a shorter wavelength. For example, at a point where U = o.75U1, the ratio of wavelengths, incident to reflected, is 9 to 1. Such short waves are unlikely to be recognized as a direct reflection without prior knowledge. Because of the substantial amplification, all wave trains eventually become too steep to be described by an infinitesimal wave theory if they meet substantial adverse currents. In practice they will break either before, at, or after reflection. Finite-amplitude effects can alter the stopping velocity. Experimental results for hydraulic breakwaters (e.g., Evans, 1955, Fig. 5) indicate an appreciable variation of stopping velocity with wave steepness, so that it is valuable to have some theoretical results for finite-amplitude waves. The result (2.115) w = k(c + U ) still holds for finite-amplitudewaves. In deep water we know that the dispersion relation has the form c2 = (g/k)[l +f(a2k2)].
(2.116)
Thus c2 = uo[I +f(azk2)](c
+ U).
(2.117)
where uo = g/w is the velocity of infinitesimal waves of frequency w on still water. For a given value off(a2k2), Eq. (2.117) has coincident roots for c if
u = -$ uo[l
+f(a2k2)].
(2.118)
Thus the maximum magnitude of U is given by substituting the maximum value of f(a2k2) in (2.118). This is 0.193 (Schwartz, 1974), and the corresponding minimum stopping velocity is U1 = - 0 . 2 9 8 ~ 0 .
(2.119)
By using Lighthill's approximate Lagrangian (2.69) the analysis may be carried further. A number of numerical solutions obtained from it are given by Crapper (1972). The functionf(a2k2)in (2.116) is replaced by s so that c2 = uo( 1
+ s)(c + U ) .
(2.120)
The wave action flux d Y / d k can be found explicitly in terms of s and c, and the requirement that it be constant gives 2u0(c
+ U)"(s) + c(c + U ) 2 ( C + 2U)p'(s) = 2uoc;P(s,) + c;p'(so). (2.121)
D. H. Peregrine
46
in which P(s) = s2 - s3 - s4
(2.122)
and a subscript 0 refers to conditions where U = 0. Although Eqs. (2.120) and (2.121) may be solved for the unknowns s and c, (2.121) is sufficiently complex to obscure interpretation. However, by using the argument leading to Eq. (2.118)we can investigate whether waves break before meeting a stopping velocity. That is, choose s at the stopping velocity, s1 say, then
u, = -$vo(l
+ s,),
(2.123)
c, = - 2 u , .
(2.124)
Substitution of these values in the left-hand side of Eq. (2.124)then gives an equation for the corresponding value of so, which is easily solved since for values of s1 less than or equal to that for the highest wave, so is small. Indeed, for s1 = 0.193, so = 0.00067. For small s, s = a2k2,
(2.125)
so that a. ko = 0.0258 in this case. The implication is that for greater values of a, ko ,waves must break before a stopping velocity is reached, and that for smaller values, waves may be reflected before they break. Another simple result is obtained if s, is also very small, in which case (2.121) simplifies to :s = 64s0
or
a, k , = 8
k0)1’2.
( ~ 0
(2.126)
Experimentally, it is found that the greater the still-water steepness of waves, the greater is the velocity required to stop them. This is in apparent contradiction to the above analysis. However, experimental waves have been sufficiently steep that they can be expected to break before the stopping velocity, and details of the breaking process are likely to affect the result. For example, the momentum lost from the waves in breaking is transferred to the current, and although this does not make a significant contribution to the mean current in deep water it will certainly affect the current distribution in the surface layer that directly influences the waves. Effects of a variation of mean velocity with depth are examined in Section IV. Holliday (1973) has calculated a few solutions for finite-amplitude capillary-gravity waves. These show a considerable variation from gravity waves near stopping points. This is not unexpected because of the variation of cg relative to c; however, the paper contains no discussion of this. If waves are at an angle to the current, that angle is an extra variable in the problem. Let 8 be the angle between k and U.The assumed current distribu-
Interaction of Water Waves and Currents
47
tion (2.83) introduces no asymmetry, so that only the range 0 I 8 I K need be considered. As already mentioned there is no interaction at 8 = n/2 so that the analysis proceeds most naturally by considering 0 I 8 < n/2. There is no loss of generality as long as both positive and negative values of U ( x ) are considered. The extra equation needed to determine 8 comes from the second consistency condition (2.42),which integrates immediately, since all. the wave parameters are independent of y, to give k sin 8 = m.
(2.127)
For steady wave conditions, or following waves along a ray, m is a constant. Again the frequency w = k(c
+ U cos 8)
(2.128)
is constant, and the dispersion relation may be used. The wave action flux B is also constant: the x component, which is the most relevant, is B, = pga2(u + cg cos e)/za
(2.129)
for infinitesimal waves. By limiting consideration to deep-water waves it is easy to eliminate k and c from Eq. (2.127), (2.128),and the dispersion relation to find an equation for 8: U * ( X )cos 8 = sin 8 - ( M sin q1I2.
(2.130)
In this equation U* is a dimensionless velocity,
U*(x)= U(x)m/w.
(2.131)
The velocity o / m is the phase velocity of the waves in the y direction. The constant M = gm/w2
(2.132)
and equals the value of sin 8 for the waves where U = 0. However, M may be greater than one if the waves are generated on moving water and cannot propagate onto water at rest. Once 0 has been found from Eq. (2.130) it is straightforward to find the corresponding values of k, c, and a. In order to illustrate the behavior of (2.130), a graphical method of solution is illustrated in Fig. 5. Each side of Eq. (2.130)is plotted against 8 for a number of values of M and U*.A given set of waves may be followed by looking along a line corresponding to the right-hand side of the equation. For waves generated on still water it is apparent that as they propagate onto a current with a component in the same direction, 8 increases and their
D. H.Peregrine
48
FIG. 5. The two families of lines correspond to the left- and right-hand sides ofEq. (2.130). Their intersections give values of 0 that solve the equation for the appropriate values of U* and M.
wavelength, from Eq. (2.127), also increases. The opposite happens when they propagate onto an adverse current, and there is a stopping velocity such that
u + cg cos 8 = 0.
(2.133)
In Fig. 5 this corresponds to a double point of intersection of the two appropriate curves. The behavior near the stopping point is similar to that already described for 8 = 0. That is, if the waves are sufficiently gentle they are reflected into waves that are swept downstream. In this case the value of 8 diminishes toward zero for the reflected wave train. Alternatively the waves become sufficiently steep for nonlinear effects to become important and may break. Crapper (1972) gives the results of three numerical calculations for this problem, using Lighthill's approximation. No unexpected behavior is apparent. There are three solutions for 8 of Eq. (2.130) in the parameter range
3$/4
> M > 1,
-&I <
U* < 0.
(2.134)
Interaction of Water Waves and Currents
49
Hence there are two ways in which a double root can arise. The new type, which only occurs in this range, corresponds to waves that are being swept downstream to a stopping point. This can happen to waves that initially are propagating almost directly across the stream. If they are swept downstream into slower moving water they are refracted until their component of c, upstream is large enough to allow
u + c,
cos 8 = 0.
(2.135)
At this stopping point they may be reflected upstream and will continue to be refracted toward the direction from which the stream comes. If the adverse current increases sufficiently upstream, a normal stopping point may be reached. A second reflection may occur and the waves are once more swept downstream. Now the crests would be nearly perpendicular to the stream and the wave number substantially increased. Figure 6 gives a sketch
- 0.3 -or -0.25
U’
extmordmary slopping pant
0.2
-0.15
-0.1
FIG.6. A sketch solution for the wave crests of a wave train, with M = 1.02, reflected at two stopping points. The arrows denote the direction of propagation of the waves (i.e., c,, not the ray direction c, + U).
of what wave crests may look like for such waves on a stream with a constant velocity gradient dU/dx. The rays do not conveniently fit on the same diagram. They are tangent to the lines of crest cusps. As is usually the case for nonzero currents, the rays are not perpendicular to the wave crests; indeed, for tan 8 = -mU/w = - U*, (2.136)
50
D . H.Peregrine
the rays are parallel to the wave crests. In Fig. 6 this occurs at a velocity just below the normal stopping velocity for the rays reflected at that point. The preceding analysis is clearly relevant to waves on rivers and flows in channels. Waves on such flows are usually generated either by wind, by ships or boats, or by obstacles. The latter waves are stationary and this a special case, with o = 0. The other two generating mechanisms generally form waves directly on the current and thus not necessarily with the parameters relevant to still-water conditions. The effects near the stopping velocity are readily observed, and result in steep waves as the theory indicates. An account of waves generated by a launch becoming steeper as they propagate upstream and hence giving an unusual sequence of events to an observer is given by Peregrine (1972). For example, a launch traveling upstream at constant velocity relative to the water, against a current with velocity increasing with distance upstream, generates waves. To an observer stationed at a point below the stopping velocity, the waves appear to increase with time after the boat’s passage. This is easily explained by the theory at the beginning of this subsection. For sea waves, the group velocity of significant waves is usually much greater than current velocities except for regions of especially strong currents near coasts. The effects of such currents producing tidal races at headlands have already been mentioned. Other strong currents are found at the entrance to harbors, rivers, and enclosed bays or lagoons. Again the effects of currents on incident waves, lengthening or diminishing their wavelength, is often clear. Johnson (1947) gives two aerial photographs showing the effect of an opposing and a helping current. Laboratory experiments to investigate the phenomenon, reported by Hales and Herbich (1972), showed strong effects. However, their photographs indicate that the current was too narrow for a theory based on large-scale current variations to be applicable. This also appears to be the case for rip currents, which occur near beaches. Rip currents are the strongest of the currents generated in, and confined to, the region just outside and within the surf zone on beaches. They are jetlike currents, usually returning toward the sea the mass flow associated with the incident waves. Quite often their position is stabilized by minor beach features, such as longshore bars, but they can and do occur on uniform beaches. There are a number of papers trying to account for the generation of rip currents, especially those on uniform beaches. For example, Noda (1974) shows how the interaction of waves and minor beach topography gives rise to strong rip currents; in a following paper (Noda et al. 1974) a partially successful attempt to include the effects of the currents on the waves demonstrates how important this is. Arthur (1950) gives a computed ray diagram showing the effect of a current over a uniformly
Interaction of Water Waves and Currents
51
sloping bottom. Le Blond and Tang (1974) use a more complete approach, including the whole cycle: incident waves generate a current that in turn influences the waves. Their method is to look for a perturbation to a steady two-dimensional solution. However, the width of rip currents is rarely much greater than the wavelength of the dominant incident waves. Thus, as Arthur (1950)acknowledges, the assumption of a slowly varying wave train is inappropriate. Indeed, whereas one might expect a more rapid increase in incident wave amplitude on the rip current, observations indicate appreciably less growth in wave amplitude. For example, a descriptive paper by Shepard et al. (1941) comments that a gap in the breakers often occurs at the site of a rip current. As this subsection shows, it is possible for the waves to be reflected by the current, but this seems a little unlikely. The results of Section II,E indicate that the lateral variation of current ought to concentrate the wave energy near the current’s center. It seems more likely that the waves are diffracted away by the current and a completely different approach is required (e.g., see Section 111). In practice the currents are usually unsteady, and the bed topography also needs to be considered. Although the more dramatic effects of the interaction between waves and currents require relatively strong currents, the more usual currents of seas and oceans do exert an appreciable influence on waves. For such weaker currents, analysis is simplified by assuming all changes to be small and then using the differentials of Eqs. (2.89) and (2.92). For deep-water waves this results in
da/2a = dk/k = - 2 dc/c = dU/(U dala = -(3U
+ fc),
+ 2c) dUl(2U + c)’.
(2.137) (2.138)
Explicit results like this may also be found when the depth of water influences the wave’s velocity. This type of approach is particularly appropriate to currents associated with long traveling waves of the form (2.84). The most commonly encountered long shallow-water waves are the tides. The above formulas can easily be used where waves and tidal current are approximately unidirectional, even if the tide is not simply a progressive wave. A standing wave component in the tide can be written as the sum of two progressive waves, e.g., 2A cos K X cos Rt = A cos K ( x - C t ) + A cos K ( x + Ct),
(2.139)
where C = a/K,and the contribution due to each of these is added together. The velocity U in (2.137)and (2.138) is equal to minus the phase velocity of the long wave, and dU equals the variation in velocity. This case of waves on
52
D. H. Peregrine
tidal currents is examined in detail by Vincent (1975). It is readily shown that wave amplitudes are amplified most at high tide, for unidirectional flows. Vincent shows that statistics derived from wave measurements in the southern North Sea are consistent with this result. For long internal waves results (2.137) and (2.138) still hold, but there is an interesting complication. Although the currents involved are weak, the phase velocity C of an internal wave is such that it is possible for
c=c*.
(2.140)
That is, effects associated with stopping velocities may occur, for surface waves traveling in the same direction as an internal wave. Gargett and Hughes (1972) report regular surface markings in a region of strong internal wave activity, which on closer inspection are regions of steep, long-crested, short waves. The paper contains two photographs of them as well as a theoretical analysis. The various stopping velocities for waves at an angle to the current are identified, and their importance is discussed. However, their physical nature and detailed solutions are not found. The waves look as if they are associated with a stopping velocity. Experiments on long internal waves with short gravity waves propagating in the same direction are reported by Lewis et al. (1974). They particularly investigated conditions near the stopping velocity. In the frame of reference moving with the internal waves, the conditions of the laboratory experiment, where both wave trains are generated at the same point, do not correspond to a steady state, so the analysis of this subsection does not directly apply. The paper presents an analysis of a perturbation about the basic state. This indicates that the situation in which the group velocity of the surface waves equals the phase velocity of the internal waves may be considered a resonant interaction. Detailed measurements are presented that are in good agreement with the theoretical results. For many purposes sea waves are best considered in terms of their energy spectra, and the transformation of spectra by currents has been discussed by Phillips (1966, p. 60),Huang et al. (1972), Vincent (1975), and Gargett and Hughes (1972), although the latter is only a brief discussion in connection with internal waves. Huang et al. (1972) give the most extensive discussion. They take an empirical form of energy spectrum E(o)for wind waves, assume that the waves are generated in still water, and then use relationships equivalent to (2.91) and (2.92) to calculate the corresponding energy spectrum after propagation onto a current. For adverse currents this results in a cutoff at high frequencies corresponding to those waves which cannot propagate upstream. For waves that are being actively generated by the wind, these higher frequencies will be in the “saturated part of the spectrum (Phillips,
Interaction of Water Waves and Currents
53
1966, Sect. 4.5). On the other hand if the waves propagate onto a current traveling in the same direction this part of the spectrum will be appreciably diminished. Huang et al. suggest that since the high wave number part of the spectrum contributes most to surface roughness, this may be used as the basis for a method of measuring surface currents. They fail to note, as Phillips (1966) points out, that an adverse current will keep the spectrum saturated, so that significant changes will be essentially due to the change in current from its maximum adverse value. For ocean currents there is the added complication that waves are generated by the wind, not only in regions of no current but also where there are currents. This is aggravated by the variable nature of most currents since the time history of both waves and currents becomes important, for example, in Barber’s (1949) calculation of the changes in wave period, due to tides, of swell from distant storms crossing the area of the continental shelf southwest of Britain. Some idea of the practical importance of such changes may be obtained from Tung and Huang (1973), although a number of simplifying assumptions are made in their analysis, which is a sequel to Huang et al. (1972).A “force” spectrum is deduced for the forces exerted on an obstacle fixed to the sea bed, e.g., a coastal structure or oil rig. For example, using a wave spectrum corresponding to a generating wind speed of 18 m sec- (40mph), their calculations show (in their Fig. 4) that an adverse current of 1 m sec(3.3 ft sec-l) increases the force spectrum maximum by over three times, whereas a simple superposition of waves and current doubles this maximum. The angular spread of a wave spectrum is also an important parameter. Inspection of Fig. 5 shows that an adverse current U tends to concentrate the wave spectrum around the - U direction, while a favorable current tends to widen the angular spread for waves generated on stili water.
E. STEADY CURRENT, VARYINGACROSS THE STREAM
A current of the form (2.141)
is the simplest form of shear flow. For any function U ( y ) it is a solution of the inviscid equations of motion for water bounded by a horizontal free surface and a bottom with depth variation h(y).The real flows that support gravity waves are turbulent, but even so form (2.141) can be a reasonable approximation in appreciable portions of the flow field. The results derived from this simple form can be used to interpret real flows. This discussion is confined to steady wave trains on flows of deep water.
54
D. H. Peregrine
If 8 is the angle between the wave number vector and the current, we again have that w=a+kcosOU
(2.142)
is constant. The consistency condition (2.42) gives
k cos 8 = I,
(2.143)
another constant, and the y component of wave action flux gives a third constant, By = pga’c, sin 9/20,
(2.144)
which with the dispersion equation is sufficient to determine the waves. Equations (2.142) and (2.143) immediately give 0
= 0 - IU.
(2.145)
k and c follow from the dispersion relation, and substitution for k in (2.143) gives
cos e = gi/(w - I U ) ~ .
(2.146)
Clearly, for a range of values of U this expression can have a magnitude greater than one. For those velocities there are no waves with parameters o and 1. The critical velocities bounding the region without waves are (2.147)
At this velocity 8 = 0 or R, so the waves travel parallel to the current. From the equations for the rays dx/dt =
U ( y )+ cg cos 8,
dy/dt = cg sin 8,
(2.148)
it is easy to show that they generally have nonzero curvature at the critical velocity. Thus the rays are tangent to a caustic line at such points. The conservation of wave action gives a’ = 8lB,/pg sin 28.
(2.149)
The amplitude becomes unreasonably large as 8 approaches 0, 4 2 , and R. We have already noted that the values 0 and R correspond to a caustic, and hence it is a place where the approximation of a plane-wave train becomes invalid. The case 8 4 4 2 is where waves are refracted so much that the wavelength becomes very short and the small amplitude approximation is no longer valid since the waves are too steep. This latter effect shows more strongly in the expression for the wave steepness a2k2 = 4l3BY/pgsin 8 c0s3 8.
(2.150)
Interaction of Water Waves and Currents
55
FIG.7. Curves showing the amount of refraction of wave trains on a shearing current. The angle 0 is plotted against the dimensionless velocity change U*.
The behavior of wave trains can be followed graphically with the aid of Figs. 7 and 8. The amount of refraction is found from Eq. (2.146), which can be cast into the form cos e = cos eo/(i- u* COS
eo)2,
(2.151)
where the wave train initially has a phase velocity co and is traveling at an angle O0 to a stream Uo . The dimensionless change in velocity
u* = (U- Uo)/co*
(2.152)
Figure 7 shows curves giving 0 as a function of U* for a comprehensive
FIG.8. The variation in amplitude, wave number, and wave steepness for a wave train refracted by a parallel shear flow.
D.H.Peregrine
56
range of values of 8,,. The corresponding variations of amplitude, wave number, and steepness, ak, are shown in Fig. 8 on a logarithmic scale. The curves are normalized by the value of each quantity at 8 = n/6,which is the angle at which ak is least. The singularities at both ends of the range of 8 are apparent. However, reference to Fig. 7 shows that, starting with some "middling" value of 8, the change in current required to reach 8 = 0 is much less than that required to approach 42.Thus caustics may occur more often than extremely steep waves due to refraction. There are uniformly valid small-amplitude approximations including caustics. Such solutions are presented in both McKee (1974)and Peregrine and Smith (1975).McKee considers the case of a shear flow with a monotonic velocity distribution U ( y ) in the neighborhood of the caustic. Peregrine and Smith consider stationary waves (i.e., w =0) on a velocity distribution that has a maximum value. This leads to waves being trapped between two parallel caustics. In all these parallel-flow problems there is no loss of generality in considering stationary waves since the phase velocity in the x direction (w/k cos 8) is constant; thus there is always a frame of reference in uniform translation in which a wave train is stationary. These two papers differ a little in the form of solution assumed, but for the first approximation in powers of k - there is no significant difference, except that McKee (1974)allows for a depth variation h(y). The solution has the expected form in that it depends directly on an Airy function. If the caustic is taken to be at y = y, and U ( y )increases with y so that the region with waves is y c y,, then
'
4 = A(ka/c,1)'/2(dY/dy)-
Ai( - Y),
(2.153)
where for y c y,,
;IYYC
Y3/2 =-
k sin 8 dy,
(2.154)
and k, a,8, and cg take the values that may be deduced from Eqs. (2.142)and (2.143)and the dispersion relation. The asymptotic expansion of the Airy function shows that the waves are perfectly reflected and that the solution is consistent with Eq. (2.144)for the flow of wave action if A' = 8alBJpg.
(2.155)
Close to the caustic, this means that (2.153)may be written
- y,)],
= (8naB,/pg~,)'~~(c,l/2U')'~~ Ai[(212U'/cg)'/3(y
(2.156)
where all the parameters and U' = dU/dY are evaluated at the caustic.
Interaction of Water Waves and Currents
57
McKee's (1974) result incorporating the effect of varying depth is obtained by replacing U' with (2.157)
Some simplification is obtained for deep water, since at the caustic c, = 241.
(2.158)
In this case the maximum amplitude is 0.536A(2a/ U')' I 6 ,
(2.159)
and the corresponding maximum steepness is I times the same expression. Since the waves have a minimum amplitude at 8 = a/4, the maximum amplification is calculated to be 1.065(a/U)'/6,
(2.160)
and the maximum amplification of steepness from its minimum at 8 = 4 6 is 0.496(a/U')'16.
(2.161)
Clearly, large amplification near a caustic is not likely. For example, the largest values of the parameter a/U are likely to occur for sea waves, but even then it is unlikely to exceed lo6, which gives a maximum steepness less than 5 times the minimum. Thus in many instances small-amplitude theory will be valid at such caustics. The natural way to consider a caustic after reading the above is to think in terms of a stronger current refracting waves until they are propagating in the current direction and reflected. An alternative view of the same wave system is to consider waves propagating with a component of their direction being upstream. Then a less adverse current may refract them to form a caustic. This is particularly likely to happen on a river or similar stream of water where the flow has a central maximum and slackens toward the edges. When such flows are sufficiently rapid they can sustain stationary waves, and in such circumstances the two caustics, one on each side of the maximum velocity, are easily seen by a casual observer. This configuration is considered in detail by Peregrine and Smith (1975). A sketch of the rays is given in Fig. 9a. When the caustics are far apart, the analysis is similar to McKee's (1974) except for the matching of the solutions for the two caustics. This gives an eigenvalue problem for the wave number 1, which is found in the first approximation from (N
+ ;)a
=lyrk
sin 8 dy,
(2.162)
D . H.Peregrine
58
h
FIG.9. Ray diagrams for waves encountering a jetlike flow. A velocity profile is at the left-hand side ofeach diagram.(a) Waves trapped in the region of maximum velocity. (b) Waves excluded from the region of maximum velocity. (c) Waves partially reflected by a "near caustic."
in which y, and y, are the positions of the caustics and N the number of zeros of the amplitude between them. Peregrine and Smith (1975) give a further approximation and solve the case where N is small, the two caustics are close together, and the appropriate ansatz involves Hermite functions. Examination of Fig. 7 shows that if the range of 8 across the stream is relatively small (e.g., f 15") then the waves exist only on velocities very close to the maximum. Thus a line of waves trapped between caustics on a stream of water gives a very meager indication of the actual velocity distribution across the stream. The converse situation, where waves are refracted away from a stream traveling in their own direction, is examined by McKee (1975). In this case, the waves are outside the region between the two caustics; a ray diagram is sketched in Fig. 9b. If these two caustics are close together, some of the wave
Interaction of Water Waves and Currents
59
action incident on one is reflected while the rest is transmitted. The ansatz is in terms of parabolic cylinder functions (Hermite functions are a relatively simple set of these). McKee’s (1975)results indicate that the reflection coefficient decreases from unity for widely spaced caustics to 2-l” for the case when ray theory indicates that they coincide and the maximum stream velocity equals that at a caustic. It is unlikely that the reflection drops abruptly to zero for a slightly lower maximum current; thus if a ray diagram indicates a “near caustic” some reflection is likely. A near caustic is sketched in Fig. 9c. It is a line of inflection points in rays where the angle between the rays and the inflection point line is small. This may be investigated with the same analysis as McKee (1975). The diagrams in Fig. 9 also partly illustrate the behavior of waves on a current such as a river. The shear flow refracts waves that are propagating downstream out of the region of maximum velocity. On a river this increases the dissipation and scattering of such waves by the river’s banks. Conversely, waves propagating upstream are refracted toward the center of the stream and as a result suffer little scattering or dissipation. This is especially evident for wind-generated waves, which even propagate upstream around corners into reaches sheltered from the wind. Similarly in these circumstances winds opposing currents will be able to generate larger waves than comparable winds in the current direction. However, in most rivers this latter effect will be obscured by the greater effective fetch available for the upstream wind. Finite-amplitude effects may be worked out for deep-water waves, away from any caustics, by Lighthill’s approximate Lagrangian (2.69).Crapper (1972)shows the results for a few representative initial conditions but because of his computation method fails to note that there is a singularity in the solution differing from that which occurs at a caustic for infinitesimal waves. Equations (2.143)and (2.145)still hold, so that using
1 + s = a’jgk
(2.163)
instead of the linear dispersion relation gives
cos e = gI( 1 + s)/(w - 1 ~ ) ’= a(y)(1 + s).
(2.164)
Note that a(y) is the value (2.146)of cos 8 for small-amplitude waves; thus one effect of appreciable wave steepness is that 8 is diminished on a given current for the same parameters 1 and w. The relevant equation for constant wave action flux is d S / a r n = const
(2.165)
60
D. H.Peregrine
where
m = k sin 6.
(2.166)
After substitution of 9 this may be reduced to sin 6 c0s3 6 s(2 + s - 9s’ + 6s3) = 28,
(2.167)
where is a constant. Equations (2.164)and (2.167)are two equations for 8 and s. For a given wave train, 8 is constant but a is a function of y; thus if curves given by (2.164)are drawn in the (6, s) plane for a range of values of a, then the appropriate curve (2.167)immediately shows the variation ofboth s and 6. Curves for representative a and 8 are shown in Fig. 10. The velocity U
P
FIG. 10. Solutions for the wave steepness (s u a2k2)and direction of propagation of finite-amplitude deep-water waves on a shear flow. The lines a(y) = const may be identified by the value of 0 at s = 0, and use of Fig. 7 then gives the corresponding value of U(y).The lines B = const do not touch either axis, and the dashed line is where lines of the two families touch.
corresponding to different values of a can be found from Fig. 7 since a = cos 6 for zero amplitude. An immediate and striking result is that there are no solutions for velocities corresponding to sufficiently small values of 6 . Each curve /I= const touches one of the family of a curves and does not meet any of the a curves between that point and the origin. At each of these points, joined by a line in Fig. 10, dslda (and hence ds/dU and dsldy) approaches infinity, even though s is finite and may be quite small. The solution for s as a function of U has two branches, which in the (s, U )plane are represented by a single smooth curve. This means that for given o,1, and U there may be two solutions (s, 6). The conclusion to be drawn from the singularity of the s derivatives is that the plane-wave assumption does not hold. There is a caustic, but the singu-
Interaction of Water Waves and Currents
61
larity differs from that for infinitesimal waves. Observation of shear flows does not reveal any tendency for waves to become unduly steep near caustics, so it is likely that the reflected waves generally correspond to solutions on the lower branch of the s, U curve. The difference from the infinitesimal case is quite striking. For example, for s = 0.11 and a velocity that causes infinitesimal waves to be at 30”to the current, the finite-amplitude solution stops with 8 = 16”. The difference is still there as s + 0. The line of double points is (2.168)
near the origin. Thus, given a sufficiently small value of dU/dy, even the gentlest waves may meet this singularity an appreciable distance from the zero-amplitude caustic position. However, one is reassured by the existence of uniformly valid approximations for infinitesimal waves reflecting at a caustic. This singularity is not an artifact of Lighthill’s approximate Lagrangian since it occurs for arbitrarily small s for which any finite-amplitudeapproximation should give the same result. It is also very similar to a singularity in s arising from an initial-value problem investigated by Lighthill (1965, Sects. 13-15). It seems likely that a similar singularity may be found in caustics caused by varying depth. In that case there is an exact linearized solution for edge waves, which provides a starting point for a uniform finite-amplitude approximation, and, hence, a possible way of gaining further understanding of the problem. There has been little experimental work on this problem. One of the major problems in experiments is to set up a uniform current with shear since most large-scale flows are turbulent and not unidirectional. Hughes and Stewart (1961) measured the propagation of capillary-gravity waves across a shear flow. Their elegant approach to the turbulence problem was to set up a stable Couette flow. The measurements of wave slope confirm that energy is not conserved and the results are consistent with inclusion of the effects of radiation stress. [Longuet-Higgins and Stewart (1964) point out that the effects of radiation stress are underestimated since an incorrect expression was used.] An interesting application of refraction analysis is given by Kenyon (1971). The propagation of waves across the Pacific Ocean from storms in the Antarctic Ocean was measured by both Munk et al. (1963) and Snodgrass et al. (1966). In the former paper, it is noted that “wave inferred directions [to storm centers] are to the left of the location from weather maps,” and in the latter paper, measurements were obtained “though the stations are totally shadowed.” Kenyon suggests, and gives some detailed
62
D. H.Peregrine
figures, that the refraction of waves by the Antarctic circumpolar current is sufficient to account for these anomalies. Abnormally large waves off the southeast coast of South Africa have been reported over a number of years, but with increasing frequency over the last decade. For example after meeting a large wave the 28,000-ton tanker World Glory broke in two in 1968,and the 260,000-ton ore/oil tanker Svealand was severely damaged by a wave in 1973. Some of these incidents are reported in Marine Observer published by the Meteorological Office, and Mallory (1974) gives details of several incidents together with meteorological and hydrographic information. Mallory's report shows that these exceptional wave conditions occur off an almost straight coastline stretching in a southwesterly direction, with a relatively narrow continental shelf of between 10 and 30 km width. The Agulhas current runs in a southwesterly direction, bounded by the outer edge of the shelf (200-m contour), with its maximum current of over 2 m sec-' just seaward of the shelf edge. Its width is 95-160 km, but it does not usually flow on the shelf, where a countercurrent of up to 1 m sec-' flows when a cold front passes. The abnormal waves are all observed on the Agulhas current, and their occurrence is either coincident with or a few hours after the passage of a cold front through the area. The southwesterly winds behind the cold front have a fetch of over 1200 km and have been blowing for more than 24 hours. The waves generated in this long fetch will have developed on the westwardflowing Antarctic circumpolar current, but meet the Agulhas current head on, resulting in an increase in amplitude of around 25 % on the linear theory for uniform wave trains. A further increase in amplitude is to be expected because of the jetlike nature of the current. Waves are concentrated by refraction onto the region of maximum velocity and most of the wave energy is confined to this region by bounding caustics. This fits in well with the observations reported by Mallory of higher waves outside the continental shelf on the current. Thus the current and wind systems combine to give a very high level of wave activity. The abnormal waves encountered have usually been much greater in amplitude and length than the general level. However, in any sea the largest waves occur with small probability. One aspect of caustics that needs investigating in this context is the behavior of a short group of waves traversing a caustic region. This may be a suitable way of modeling the real sea waves, which are rarely coherent for more than a few wavelengths. It seems likely that a solution would show just one or two large waves persisting for a limited time. Ocean current and wave systems of this sort merit further study since it should prove possible to forecast them and advise shipping accordingly.
Interaction of Water Waves and Currents
63
Also such waves may occur elsewhere in similar conditions; Casey (1974) in a brief letter mentions an incident off Ushant, where tides would cause the currents, and off Japan. WITH F. FLOWS
SIGNIFICANT VERTICAL
ACCELERATIONS
The foregoing theory does not apply directly to flows in which the water has appreciable vertical acceleration or surface slopes. Examples of this type of flow are surface gravity waves on deep water and the flow over a waterfall or weir. The former example is of most interest, and the behavior of short waves riding on long waves has been discussed by several authors; it was in Longuet-Higgins and Stewart’s (1960) paper on this subject that the concept of radiation stress was introduced. In that paper a careful perturbation analysis for short waves riding on long waves is presented. Both sets of waves are assumed to have small amplitude and the interactidn occurs in second-order terms. Once the appropriate amplitude variation is described, a physical interpretation is discussed, including among other things the idea of radiation stress. It is seen that the effect of the vertical acceleration of the water on the dispersion relation needs to be taken into account. At first sight this appears inconsistent, since all other derivatives of the long-wave velocity field are ignored. In that example, a confirmatory check is available from the perturbation analysis. A more general illustration of the effect of vertical acceleration is given here. For simplicity of presentation consider the basic flow to be steady and two dimensional; this includes periodic plane waves that are steady in a frame of reference moving with their phase velocity. Introduce arthogonal curvilinear coordinates (s, n) near the water surface such that s is measured along the surface and n increases along normals outward from the surface. Let the basic flow have (s, n) velocity components (U, V ) and pressure P satisfying the equations of motion. Then the boundary is n = 0, and the boundary conditions are V=O and P =const. (2.169) This flow together with infinitesimal waves riding on it may be described by velocity components (U + u, V + u), a pressure P + p , and the free surface n = q. These satisfy linearized inviscid equations of motion: au -+-at
(1
a ( U u ) + - a( V u ) + K U V + K U u +
+ m)a s
an
p( 1
= 0, + m)2 as
(2.170)
D. H. Peregrine
64
together with linearized boundary conditions applied at n = 0
2 +--- u ar7 - v, at
(1
(2.172)
+ K n ) as
ap p = --q. an
(2.173)
In these equations K is the curvature of the free surface of the basic flow, with K > 0 if the center of curvature is within the fluid. The equations of motion for the basic flow give
-
-ap/an = p v p t ) - g ii,
(2.174)
where g is the gravitational field and ii the unit vector normal to the free surface. Thus, taking (2.169) into account the boundary condition (2.173) may be rewritten p =(-g
a - K u2)q.
(2.175)
Flow along the surface streamline will satisfy Bernoulli's equation so that U is of the order (gL)1'2,where L represents the length scale of the basic flow. This is the case for deep-water long waves since U is then approximately equal to the phase velocity, which is ( g L / 2 ~ ) 'where / ~ , L is the wavelength. If the radius of curvature K - ' is also of order L, then the terms uU2 and g ii are of the same order of magnitude and should both be included in the first approximation. For a small-amplitude deep-water long wave K - is of order AIL2, where A is the wave amplitude, and KU' is of order (A/L)g ii. For small AIL it might be assumed negligible. However, this is nat the case since the variations in U that are of interest are also of order (A/L)U.Thus the term K U ~ should again be included. Typical terms in the equations of motion are
-
-
For these terms, the primary assumption that the waves are short compared with L clearly indicates that velocity gradients and the curvature of streamlines need not be included in a first approximation. Thus the dispersion relation for short waves is a2 = (o- kU)2 z
(-g
*
ii
+ DV/Dt)k = g*k,
(2.176)
Interaction of Water Waves and Currents
65
where g* is the “effective gravity” in a frame of reference moving with the free surface of the large-scale flow. For a steady situation it is reasonable to suppose that wave action is conserved and that the wave action flux
B = E( U + cg)/c
(2.177)
E = 4pg*aZ.
(2.178)
is constant, where However, as Bretherton and Garrett (1968) point out, there is ambiguity possible in defining E, the “perturbation energy density” in moving reference frames. This is made apparent by Longuet-Higgins and Stewart (1960) where the form
E = apg*az + $pgaz
(2.179)
is chosen for waves on a basic flow with small surface slopes, the second term representing potential energy in the gravitational field only. The choice (2.178) seems more appropriate since it corresponds to equipartition of kinetic and potential energy densities. This choice is also supported by consideration of the Lagrangian for a perturbed flow derived by Bretherton and Garrett, Eq. (4.19), although it should be noted that this involves energy per unit horizontal area rather than per unit area of mean free surface. The solution of these equations differs from that for the simpler flows of Section II,D only in the substitution of g* for g, so that most results carry over directly once g*(s) and U(s) are prescribed. However, the example of short waves riding on long waves traveling in the same direction involves a different solution from that discussed in Section II,D. In a frame of reference in which the long waves appear stationary, the current is U ( s )=
-c
cos II/
+ U*(s),
(2.180)
where C is the phase velocity of the long waves, $(s) their surface slope, and U*(s) the water velocity due to the waves. That is, the short waves in this frame of reference are meeting an adverse stream of magnitude C . The phase velocity of the short waves is much less than C, so that 0 = k(c
+ U)
(2.181)
is negative, showing that c is always less than - U ( s ) . Similarly the wave action flux B is also negative, and the solution illustrated in Fig. 4 does not represent this situation. For the case g* = g, the variation of a, k, and ak relative to their values at U = -2c are shown in Fig. 11. A large range of velocities are shown in the
66
D . H . Peregrine
/
0.1
4
FIG.11. The variation ofamplitude, wave number, and wave steepness ak on a current U ( S ) for negative values of w and E. Note both ordinates are logarithmic. The suffix 2 refers to values where U = -2c.
figure, since for a large amplitude long wave, - U ( s )varies from a value greater than C in the trough to almost zero near the crest. The relatively rapid variation of steepness a k of the short wave with velocity U(s)is clear. The solutions shown in Figs. 4 and 11 can also be used when g* differs from g. Let
Y = g*/go
(2.182)
7
where go is the value of g* at a point where values of the wave parameters are known. Introduce new variables c
k* = k/y,
c* = yc,
a* = a/y,
U* = yU.
(2.183)
Then the equations to be solved for the starred variables are those for constant g* = g o . The simplest example with significant vertical acceleration is where the long waves are of infinitesimal amplitude with surface displacement A sin(Kx - Of).
Interaction of Water Waves and Currents
67
Then
U ( S )1: - C
+ AR Wth Kh sin Ks,
g*(s) 1: g - ARZ sin Ks.
(2.184) (2.185)
Since A is small the analysis is simplified by assuming U = -C
+ dU
and
g*(s) = g
+ dg*,
(2.186)
and using the differentials of Eq. (2.177) and (2.181) and the dispersion relation. The results of this analysis, after identifying dU and dg* with the appropriate terms in (2.184) and (2.185), and after simplifying even further by neglecting c compared with C, are
da/o = iAK(coth K h - tanh Kh) sin Ks,
(2.187)
dklk = AK coth Kh sin Ks,
(2.188)
da/a = $AK(3 coth Kh + tanh Kh) sin Ks,
(2.189)
d(ak)/ak = tAK(7 coth Kh + tanh Kh) sin Ks.
(2.190)
Longuet-Higgins and Stewart (1960) derive results corresponding to and agreeing with (2.188) and (2.189) in their perturbation analysis. Vincent (1975) points out that although c c C,it is often not small enough to be neglected. There is only a simplification of algebra gained by its neglect. It is clear that short waves steepen as the crest of a long wave overtakes them. If the long wave has appreciable amplitude they may steepen sufficiently to break at some point on the forward facing slope of the long wave. If short waves travel in the opposite direction to the long waves, then U ( s ) is positive (supposing that the short-wave direction is again taken to be positive). However, Eqs. (2.187)-(2.190) still hold for infinitesimal long waves since dU/U still has the same value. It is suggested by Longuet-Higgins (1969) that the variation of steepness of short waves on longer waves may contribute to the growth or decay of the longer waves. Two mechanisms are proposed, one weak and one strong interaction. The weak effect is the viscous decay of the short waves, which is proportional to wave steepness. One result of the decay is to transfer momentum from the short waves to the “current.” Since the short waves are steeper at crests than in troughs, more momentum is transferred at the crest, leading to a growth or decay according to whether the short waves are traveling with or against the longer waves. Longuet-Higginsdiscusses details of the momentum transfer, introducing a virtual tangential stress at the surface, and makes an estimate of its value for a typical wind-driven system.
68
D. H. Peregrine
It may account for more than 10%of the stress due to the wind at low wind speeds, and less for high winds. It is also shown that
_1dA - - - - 4v (ak)2fJf23, A dt
(2.191)
g2
where v is the kinematic viscosity; the strong dependence on the frequency of the long waves is evident, but it is possible that the effect may be significant in their amplification. A stronger effect can be expected when the short waves steepen sufficiently to break. This gives a very direct transfer of momentum, and also, since much of the wave energy may be dissipated, more of the wave momentum may be transferred. If the wind can regenerate the short waves between the long-wave crests, this could be an efficient mechanism for generating waves with phase velocities greater than the wind speed. Longuet-Higgins’ (1969) estimates for the rate of growth of the waves are consistent with this mechanism being important for certain sea states. A contradictory result is obtained by Hasselmann (1971). The inviscid equations of motion are averaged and the interactions are considered from an Eulerian viewpoint. This represents the mass flow associated with the short waves as a surface flow occurring between the wave troughs and their crests. This results in a new kinematic boundary condition for the mean flow. At the mean free surface, z = Z ,
+
(dZ/dt) (U * V1)Z - W = -V1
. M/p,
(2.192)
where M is the mass flow associated with the short waves. The term on the right-hand side gives the rate at which the mean surface Z must be lowered to supply water to feed increases in the mass flow of the short waves. This mass flow is greatest down the front face of a crest of the longer wave, effectively transferring water from the crest to the trough of the wave, and hence reducing its potential energy. Hasselmann (1971) deduces that this term is as effective in damping a wave, as the momentum transfer is at amplifying it. Using infinitesimal wave theory he analyzes the residual terms in an expression for the rate of change of the long wave and deduces that, whichever way the short waves travel, the long wave is damped. In a discussion of radar measurements of short-wave spectra he concludes that the magnitude of this damping is “ of marginal significance.” The approaches in these two papers are difficult to reconcile. Clearly Hasselmann introduces an important interaction, which Longuet-Higgins was unaware of, but the momentum transfer in his inviscid model may be a poor representation. This is especially so for breaking short waves, where the vorticity generated may spread below the surface layer. Similarly, the re-
Interaction of Water Waves and Currents
69
liance on linear theory for order of magnitude estimates may not be adequate, especially as for short waves of small amplitude there is no significant energy transfer. More work is needed on this problem, but it is reasonable to conclude that Longuet-Higgins overestimates any amplification of long waves. A closely related subject is the generation of capillary waves near the crest of a steep gravity wave. These are interpreted as waves of the same phase velocity generated by the excess pressure at the sharply curved crest due to surface tension. Longuet-Higgins (1963) analyzes their generation by a perturbation scheme and discusses the results in the context of short waves on a large-scale flow. Crapper (1970) starts from this point of view and uses his solution for finite-amplitude capillary waves (Crapper, 1957) to calculate the variation in their steepness along the gravity wave profile. In this problem, as in most problems where surface tension is important, it is necessary to take account of the dissipation of the short-wave motion by viscosity. For this reason wave action is not conserved and it is more appropriate to use an energy equation, with radiation stress, dissipation, and an input function. For short waves of this type, if surface tension is dominant then both gravity and surface accelerations are unimportant and may be neglected. It is clear from the last few paragraphs that the dissipative effects of these capillary waves may be important and should be investigated further. Another mechanism for generating capillary waves on the front of a gravity wave becomes evident once the properties of capillary-gravity waves are considered. These waves have a minimum value of their group velocity. We have noted that U c is always negative for short waves traveling with longer waves, in the frame of reference with the long waves stationary. For gravity waves, this implies U c, is never zero, but this is not so for gravitycapillary waves. U cg may be zero and there is then a stopping point. The magnitude of the velocity U at the stopping point, and hence at the crest of the long wave, must be less than the wave velocity at which c = c,, which equals the minimum phase velocity. For normal gravity and clean water, this is 0.23 m sec- These reflected capillary-gravity waves are longer than those generated at the crest. This is easily seen from graphical consideration of the equations
+
+
+
’.
- kU,
(T
=
(T
= (gk
+ Tk3/p)’”.
The reflected waves have w < 0, while those generated at the crest have o = 0 (note that U < 0). A further factor influencing short waves riding on long waves on the sea is the vorticity distribution in the water due to direct wind stress and the
70
D . H. Peregrine
dissipation and breaking of waves. Shemdin (1972) gives experimental measurements of velocity profiles and estimates the effect on the dispersion relation, and Banner and Phillips (1974) discuss its effect on the maximum amplitude of the shorter waves (see Section IV). As already described, when the crest of a steep long wave catches up with short waves they increase rapidly in steepness. Dagan (1975) makes the interesting suggestion that for the highest waves, where U(s)is near zero at the crests, the rapid increase of amplitude of the short waves may be interpreted as an instability of the basic flow. That is, the initiation of breaking might be described in these terms. Two properties of the breaking process are described by this hypothesis: (i) its initiation is rapid, (ii) breaking occurs on the front face of a wave. On the other hand, breaking does not resemble an oscillatory short wave. If U ( s ) is always greater than the minimum phase velocity, then sufficiently smallamplitude wave disturbances may pass over the wave crest, while if U ( s )has a lower value, there is a stopping point near the crest where waves may be reflected, and for sufficiently small initial waves, infinitesimal wave theory may be used to find the amplification. Thus the usual requirement for instability of indefinite amplification of an arbitrarily small disturbance is not met. Dagan’s analysis is for a steady flow and makes no assumption about the rate of change of the short waves, in which respect it is valuable. It certainly indicates that short waves may sometimes precipitate or influence breaking. In confirmation of this the author has a 16-mm film showing a small wave disturbance meeting a wave on the point of breaking on a beach. The larger wave breaks with two sheets of water projected forward. By running the film backward frame by frame it is clearly seen that one of the sheets of water is directly connected with the incident disturbance. 111. Small-Scale Currents
A common way of finding solutions to difficult mathematical problems is to look for parameters that may be either very large or very small and then to solve the problem for those cases by making appropriate approximations. This approach is successfully used in Section I1 to deal with large-scale currents. At the other extreme are current distributions with a scale much smaller than a wavelength. Some important examples are best discussed in a context covering all length scales; thus flows that vary with depth are treated in the next section, and the interaction of the flow around a ship with the waves it generates is considered in Section V. This leaves few situations of interest where much analysis may be done, and most of this section is devoted to thin shear layers.
Interaction of Water Waves and Currents
71
There has been very little work on this problem, so it is of interest to note an analogous problem that has been studied more intensively.The propagation of sound through moving fluids is one such case; indeed, twodimensional sound waves in a uniform atmosphere satisfy exactly the same equations and boundary conditions as infinitesimal shallow-water waves in water of constant depth. This subject is reviewed in a paper by Lighthill (1972) and the book by Goldstein (1974) gives mathematical details of some of the topics. Naturally many of the problems that arise in acoustics have little relevance to water waves and vice versa, although the mathematical methods are applicable to both fields, or at least provide a useful starting point. When all dimensions of a current system are very small compared with those of the waves, the wave may simply be taken as giving the local mean water level and current as slowly varying functions of time. For example, this is often done in relation to the tides for small-scale coastal problems. The effect on the waves is negligible, unless there are many such small current systems. In any case, the author does not know of important or interesting examples. More interesting currents are those which have one long length scale but which are otherwise short compared with the waves. A thin shear layer between two different, nearly uniform flows is the simplest example. A thin jet is another example. In real flows regions of strong velocity gradients only remain thin if there is some factor opposing their usual turbulent spread. However, in many cases a portion of the flow field might be well described as a thin shear layer, and any solution for that case can be of value in interpreting or predicting behavior in the problem where the current scale is of the same order as that of incident waves. In searching for mathematical solutions to problems involving thin shear layers, it is a natural step to look at the limit as the thickness of the shear layer goes to zero. That is, to consider a flow with a vortex sheet across which the velocity is discontinuous. In practice both vortex sheets and thin shear layers are unstable flows, so that steady solutions to such problems cannot be expected to give more than a crude approximation to real situations. This is better than nothing and may be quite adequate in some circumstances. If a vortex sheet at y = q(x, z, t ) separates two regions of flow, denoted by subscripts 1 and 2, then the boundary conditions on the vortex sheet are (i) that the pressure is continuous, Pl
=P2
9
and (ii) the fluid particles each side of the vortex sheet move with the sheet,
72
D . H. Peregrine
that is,
art
- + u.-aY
at
+ w . -art = u i ,
lax
for i = 1, 2.
laz
This latter boundary condition has been incorrectly formulated in a number of papers, both for compressible flows and for water wave problems, by the omission of the last two terms on the left-hand side of Eq. (3.2). The simplest problem is for an undisturbed flow that consists of a plane vortex sheet with uniform flows ( U1, 0,O)and (V,, 0,O)on its two sides. The incident wave is simplest if it is plane periodic, making an angle 8, with the flow direction. Solutions for the acoustic problem were given independently by Miles (1957) and Ribner (1957). The corresponding linear shallow-water problem is quite straightforward. Matching phases on both sides of the vortex sheet gives
kl cos 8, sec 8,
= k,
cos 0 2 ,
(3.3)
+ F , = sec O2 + F, ,
(3.4) where tI2 is the angle the transmitted wave makes with the flow and Fi is the Froude number Ui/c= Ui(gh)-’”. These show that there are two critical angles, and incident waves are totally reflected if
- 1 I sec 8,
+ F , - F , I 1.
(3.5) The amplitudes of waves are easily found from the linearized boundary conditions. If A iand Biare the amplitudes of waves propagating in the + y and -y directions, respectively, on the appropriate sides of the sheet, then A1
+ B1 = A , + B 2 ,
k , sin $ , ( A , - B , ) = k , sin $ , ( A , - B2).
(3.6) (3.7)
If the signs of cos 8, and cos 8, differ, then the reflected and transmitted waves may be many times larger than the incident wave. This solution only occurs for lF,-F,l>2 (3.8) and appears to be a mathematical curiosity since the amplification is much diminished if a finite shear layer is incorporated into the mathematical model (Graham and Graham, 1969). Much further work has been done on the acoustic problem: in particular, Jones and Morgan (1972) solve for an instantaneous line source situated off the vortex sheet. After the wave produced has interacted with the vortex sheet for a finite time, an “instability wave” arises, which has an exponentially growing amplitude. Jones and Morgan (1974) use a very simple
Interaction of Water Waves and Currents
73
method of modeling the turbulence that must arise, and this leads to a more complete discussion of the scattered sound. The papers are also of interest for the mathematical techniques used to ensure that the solutions satisfy causality. For waves in deep or moderately deep water, the acoustic analogy is not available. Even the simplest problem of linear plane waves on a vortex sheet has not been solved. The difficulty arises in satisfying the boundary conditions on the vortex sheet at all depths. Matching the phase of the wave in x gives
k, cos 8, = k, cos 0,
(3.9)
again, but matching the frequencies leads now to (3.10) for deep water instead of (3.4), because of the different dispersion relation. Given k l and el, this is sufficient to determine k, and B,, and thus the range of total reflection, which again lies between two critical angles, and is given by However, kl # k,, except in one isolated case. Thus the variation of the wave motion with depth, that is, exp(k,z), is different on the two sides of the vortex sheet. A solution that includes terms whose influence is confined to the neighborhood of the vortex sheet is needed. If the velocity potentials are assumed to vary like exp{i(lx - wt)}, where 1 = ki cos Bi ,
(3.12)
then the remaining problem looks deceptively simple and symmetrical. The equations to be solved are (a24i/ay2)
+ (a2+i/az2)
-124= ~
o
(3.13)
in the region y I0, z I0, if i = 1, and in y 2 0, z I 0, if i = 2. The boundary conditions on z = 0 are
a4i/az = a ; 4 i ,
(3.14)
and those on y = 0 are (3.15)
74
D. H. Peregrine
where ai =
10
- lUi lg-1’2.
(3.16)
Those who are mathematically inclined may like to prove the existence, or nonexistence, of solutions, or find some. Evans (1975) has succeeded in reducing the problem to that of solving a pair of singular integral equations. Conservation of wave action is proven and approximate solutions are found. Figures 12 and 13 show the results for four angles of incidence. The reflection and transmission coefficients given are simply the ratio of the surface amplitudes in the relevant waves. It may be noted that unless 8 is small the reflection is low except in the vicinity of critical angles, or when there is total reflection. When there is little reflection the transmission coefficient differs little from that for a wide shear layer for which results are given in Section II,E and shown on Fig. 12. Evans used two different approximations; both are shown in Fig. 13 and thus give an idea of the accuracy that may be expected. For
I u, - u2 1 s c1,
(3.17)
it appears to be quite adequate for any application. Another solution involving vortex sheets is given by Peregrine and Smith (1975). The solution is for stationary waves on a “ top-hat ” jet. This type of solution is relatively simple since, on introducing a velocity potential within the jet and noting that there can be no motion outside it, the boundary condition (3.1) on the bounding vortex sheet reduces to
a4lax = 0,
(3.18)
while (3.2) becomes an equation for finding its displacement. For a rectangular jet of width b and depth h, surface waves of the form
5 = a sin(nxy/b) cos lx
(3.19)
gk = U212tanh kh,
(3.20)
have a dispersion relation where
k2 = 1’
+ n2n2/b2.
(3.21)
If the currents are taken to be as weak as the water velocities in the wave motion, then it is appropriate to make a perturbation expansion with the first approximation being a simple superposition of the two (as mentioned in Section II,C,3).
Interaction of Water Waves and Currents
75
3.0 5"
'\
2.5
2 .o
0.5
-3
-2
0
-1
1
2
[u, - u,)/c,
FIG. 12. The modulus 17'1 of the transmission coefficient for waves of unit amplitude incident on a vortex sheet (solid lines) compared with the transmission coefficient for a slowly varying change of velocity (dashed lines). The angle O1 is the angle between the crests of the incident waves and the current and differs from the angle O1 used in the text. (From Evans, 1975, Fig. 1.)
75"
0.6
(u, - u, )/c, FIG. 13. The modulus of the reflection coefficient for waves of unit amplitude incident on a vortex sheet, two different approximations, solid lines and dashed lines. The angle O1 is the angle between the crests of the incident waves and the current and differs from the angle O1 used in the text. (From Evans, 1975, Fig. 2.)
76
D . H.Peregrine
IV. Currents Varying with Depth
A. INTRODUCTION There are two major causes of steady currents that vary with depth. Wind stresses at the surface and frictional stresses acting on the bottom. Viscous stresses and turbulent Reynold’s stresses transmit these to the body of the flow, setting up a mean velocity profile. One class of such flows are those where viscosity and surface tension are important. These may be described as thin-film flows and are of great importance in chemical engineering and hence have a substantial literature, both theoretical and experimental. No more than occasional reference is made to those flows here since they are outside the scope of this paper. For two-dimensional high-Reynolds-number flow in a stream with a free surface, the velocity profile is often taken to have the form U ( Z )= A Z ” ~ , (44 although measurements indicate a maximum velocity below the free surface (this may be due to surface stress from still air or to three-dimensional effects). Near a rigid bottom, z = - h, the velocity may be better represented by the “law of the wall” logarithmic profile U ( z ) = ( U * / K ) log[(z
+Wol.
(4.2) Similarly, near the free surface, z = 0, the wind-induced current may be described by
Shemdin (1972) reports measurements from a wind/wave flume and shows that they are in reasonable agreement with this formula. Wu (1975) also reports measurements from a flume that indicate a linear variation of velocity in the top few millimeters, that is, a laminar sublayer. In many cases the major part of the velocity variation is confined to boundary layers at the surface and on the bottom. Then the wind drift will only directly affect the shortest waves and only long waves, influenced by the whole velocity profile, will be affected by the bottom boundary layer. Thus a whole range of “ intermediate” length waves will be only slightly influenced by the velocity variations. Two other causes of flows varying with depth are (i) density stratification, which can lead to internal waves and to “selective withdrawal” from a stratified reservoir, and (ii) sudden increases in depth of river beds or
Interaction of Water Waves and Currents
77
artificial channels with a resulting separation of the flow such that in extreme cases it may form a surface jet. Hydraulic breakwaters also take the form of a fast surface current. In the rest of this section it is assumed that the current is in one direction only. Waves may be at an angle to the current. However, this threedimensional problem is not discussed in most examples since if a twodimensional solution is known then a corresponding solution at an angle 8 to the current is readily found, as indicated below, following Benney (1966). The x axis is chosen in the direction of wave propagation so that the basic flow is
( V ( Z )cos 8, U ( Z )sin 8, 0),
(4.4)
and the wave motion depends on x and z only. The momentum equations in the x and z directions, the continuity equation, and the boundary conditions are then exactly the same as for two-dimensional waves on the flow V ( z )cos 8 in the direction of their propagation. (That is, as long as the pressure boundary condition is not rewritten by using a form of Bernoulli’s equation.) This is because the motion is independent of y, and the velocity components in the y direction only occur in the y momentum equation, which thus becomes an equation for finding the y component of velocity once the rest of the problem is solved. This is true for finite-amplitude waves, but in that case can only apply for a single wave train or for two wave trains traveling in exactly opposite directions. B. INFINITESIMAL WAVES
1. Equations for Variation of Wave Motion with Depth Taking a basic velocity field
u = (V(z),0, 0)
(4.5)
and the inviscid equations of motion, linearized equations for a perturbation may be written. If the depth is assumed constant with the bottom of the flow at z = - h, the perturbation quantities may be taken to vary like
f (z) exp{i(kx - w t ) }
(4.6)
without loss of generality. It is then straightforward to eliminate all but one variable, giving a second-order equation for its z variation. If pressure is chosen, the equation is P”(z) - [2U’/(U- c ) ] ~ ’ ( z-) k 2 p ( z ) = 0,
(4.7)
where a prime denotes a derivative with respect to z, and c = o / k . Another
D. H . Peregrine
78
convenient variable is the vertical velocity, which gives the alternative equation W”
(
L)
- k2 + _ _ w = 0.
This equation is the “ inviscid Orr-Sommerfeld equation” or “ Rayleigh equation ” of hydrodynamic stability theory. Clearly, if the velocity profile is such that it is unstable, the unstable perturbations are possible solutions as well as solutions corresponding to periodic surface waves. Stability is discussed in Section IV,D. A full discussion of this equation, in the context of the stability of flows with rigid boundaries, is given by Drazin and Howard (1966) and a shorter account may be found in Yih (1969, Ch. 9, Sect. 6). At a rigid bottom z = -h, the boundary conditions for these equations are p’/( U
- c) = 0,
(4-9)
w = 0.
(4.10)
At the mean free surface z = 0, the linearized boundary conditions are gp’ = k2( U - c ) ~ P ,
(U - C y w ’ = [ g
+ ( U - c)U’]w.
(4.1 1) (4.12)
Equations (4.7) and (4.8) can only be solved explicitly for a few simple functions U(z),so that it is often useful to consider composite profiles. The matching conditions at a discontinuity of velocity and/or velocity gradient are that either p
and p’/(U - c)’
(4.13)
or w/(U - c) and ( U - c)w’ - U’w
(4.14)
be continuous. Occasionally continuity of w’ has been wrongly used at discontinuities of ve1ocity.gradient instead of the second of (4.14). For general wave numbers and frequencies, analytic solutions are only available for uniform currents and for currents depending linearly on z. Solutions for w are easily found and p is obtained from the relation k 2 p = U’W- ( U - c)w‘.
(4.15)
Crude approximations to most velocity profiles may be made with two or more linear regions. Although analytic dispersion relations are found, even a bilinear profile leads to complicated relations that take some effort to interpret. A number of authors have used linear and bilinear velocity profiles in
Interaction of Water Waves and Currents
79
different circumstances. For example, Taylor (1955)finds stopping velocities for a hydraulic breakwater, and Betts (1970) studies instabilities in a flume where the flow’emerges from a closed section. For stationary waves, c = 0, analytic solutions may be found for a wider range of velocity profiles. Peregrine and Smith (1975) give a short table of solutions for various jetlike flows over still water, and corresponding solutions for finite depth are possible. Lighthill (1953) gives the solution for U ( z )= Uo(h
+ z):
(4.16)
and Freds~e(1974) uses the velocity profile U ( z )= U1 cos p(z - z o )
(4.17)
very effectively to model stream flow over an obstacle. There is no intrinsic difficulty in numerical integration. Fenton (1973) gives a method that is appropriate when an analytic expression is available for U(z),but it may need modification if tabulated values of V ( z )are used since it involves v”(z). Fenton presents full dispersion diagrams showing c as a function of stream velocity for a wide range of wave numbers for the simple linear profile and the one-seventh power profile (4.1). Numerical integration is also used by Shemdin (1972). For wind waves, he takes the profile (4.3) together with the corresponding velocity profile in the air. The results for the phase velocity of short waves are in agreement with his experiments.
2. A Particular Class of Velocity Profiles One can give a.picture of the effect of different velocity profiles on waves by considering the solutions for stationary waves on flows satisfying U f f= aU,
(4.18)
where a is a constant. That is, U ( z )= U o cosh al/’z
+ Voa-‘I’ sinh al/’z,
(4.19)
where a may be positive, zero, or negative. There are three disposable constants a, U o, and Vo, so that this form can be used as a rough approximation to a variety of flows, and, although only stationary waves are considered, traveling waves can be included if c is supposed known. The solution for w ( z )for the profile (4.19),with a rigid bottom at z = - h, is W ( Z ) = sinh{(k’
+ LY)~/’(z+ h)},
(4.20)
D. H.Peregrine
80
and the surface boundary conditions give the dispersion relation
+
+
(k2 + a)’’’h coth{(k2 a)’/’h} = ( g h / U i ) ( Uoh/Uo).
(4.21)
If k2 + a is negative, this relation still holds, noting that coth ix = - i cot x.
(4.22)
The corresponding dispersion relation for a uniform flow kh coth kh = gh/Ug
(4.23)
Z = gh/Ug,
(4.24)
is included. Introducing the inverse of a Froude number squared, we consider the two relations (4.21) and (4.23) as functions of (2, k2h2).In the (2, k2h2)plane the curve (4.21)is identical to (4.23) if the latter is displaced (- Ub h / U , , -ah2), or more conveniently, if a graph of (4.23) is given, then a new origin is chosen at the point (Uoh / U o , ah2) to give the curve (4.21). Figure 14 shows Eq. (4.23) on the (Z, k2h2)plane. One advantage of this plot is that local, exponentially decaying surface disturbances are also included, in the region k2 c 0. The curve has other branches, for k2h2 < -n2, which do not appear in the diagram. Such disturbances are needed to describe fully flow near an obstacle or wave-generating object. Figure 15 is a complementary diagram showing velocity profiles corresponding to different choices of origin. The effects of velocity gradient and curvature in determining the position of the curve are clear. The origin is on the curve if V (- h) is zero, and if the velocity profile has a zero above the bottom the corresponding origin is on the right-hand side of the dispersion curve. These cases are discussed below. An advantage of these velocity profiles is that Eq. (4.8) has constant coefficients and it is thus straightforward, in principle, to use Fourier transforms to find solutions. Fredsere (1974) makes use of this. He chooses an appropriate “ truncated cosine ” profile to model the stream flow and proceeds to calculate details of the stationary waves formed by a small oblique ridge on the bottom of a wide stream. The theory agrees well with experimental results, which Fredsere presents for the variation of wave number with Froude number. Another example is calculated corresponding to a rounded bulge in the middle of a channel of finite width. The results of these two calculations are shown in various ways. When compared with a uniform flow with the same surface velocity they exhibit a reduction in surface amplitudes. However, the paths of fluid particles at the bottom of the flow show that their transverse displacement is considerably larger (e.g., twice as much) in the shear flow than it is in irrotational flow. A
Interaction of Water Waves and Currents
81
-8
FIG.14. The dispersion relation for stationary waves on a uniform stream. Z = gh/Vi
f 1,-11
f 0. -11
fl,-fl
FIG.15. The velocity profiles corresponding to the dispersion relation given by choosing the point indicated as a new origin in Fig. 14.
D. H. Peregrine
82
rough physical interpretation is given. To first order the fluid moves with the velocity appropriate to its level in the flow. Thus if its path has curvature K, a horizontal pressure gradient of magnitude pxU2 is needed. However, the perturbation pressures are unlikely to vary in the same way as V’(z), so the curvature must vary. For the cases where perturbation pressures vary slowly compared with U 2 ( z )the curvature must increase relatively rapidly to balance any marked reduction in V ( z ) .(Compare with boundary layer theory, where pressure is taken as constant through the layer.) A more mathematical way of looking at this particular phenomenon is to note that Eqs. (4.7) and (4.8) have a singular point just below the bottom, in the cases studied by Fredsere (1974). 3. A Critical Layer in the Flow
A singular point of Eqs. (4.7) and (4.8) occurs at z = z1 if V(zl) - c = 0.
(4.25)
If z = z l is in the fluid, this means that there is a critical layer at that level. The solutions in the neighborhood of a singular point, for sufficientlydifferentiable U(z),may be found by expanding V ( z )in a Taylor series in
z = z - 21. As is well known from the theory of second-order differential equations, one solution is always regular and the other may be singular. In Fredsere’s example, both solutions for p and w are regular but u and u have a singular solution. If part of the singular solution is needed to satisfy the boundary condition at the bottom, u and u could be large without there being a critical layer actually in the flow. It is instructive to look at the general solution for an oblique wave exp{i(lx + my - ot))near a critical layer. In this case IU(Zl) - 0 = 0,
(4.26) (4.27)
The results are = B[ 1
- + k 2 z 2- 2k2(v; / v ; ) zlog ~1 z 11 + A Z +~ 0(z4log I z I ), (4.28)
Interaction of Water Waves and Currents
+-1’(u;)2(31’ + m’,
83
ml, -ik2Ul/U‘;) (4.29)
where A, B are constants multiplying the regular and singular solutions, respectively, and kZ = 1’ + m’. The most striking aspect of this result is that the most singular behavior is in the perturbation velocity in a direction perpendicular to the wave number vector of the wave train. This Z - ’ term goes to zero for waves traveling against the current since its variation with 8, the angle between k and U,is sin 8 sec’ 8.This becomes very large for 8 near 4 2 , but the possibility of a critical layer then is remote because condition (4.27) becomes dificult to satisfy at a depth where the effects of the wave motion are significant. The singular solution cannot be used directly as a description of wave motion. At a critical layer other properties of the flow, neglected in the present analysis, must be introduced in order to find a physically sensible description. This aspect of critical layers is extensively studied in the theory of hydrodynamic stability. The usual method of proceeding is to include the effects of viscosity, which are used to get a solution valid in the neighborhood of the critical layer. This may be matched with an inviscid solution each side of a layer. This is relatively straightforward for unstable, growing modes of which there are usually only a finite number. The inclusion of viscosity also introduces a set of damped modes. For the water wave problem, Craik (1968) presents an analysis for resonant interactions among a triad of waves on flow with a uniform shear, and Velthuizen and van Wijngaarden (1969) consider long waves in a channel and attempt to find their rate of decay. Velthuizen and van Wijngaarden are concerned about the problem of upstream propagation against fast flows (see Section IV,B,S) so they assume a critical layer for very long waves even though there is a solution without a critical layer. However, a full discussion of solutions with critical layers should take into account how the waves may be generated, and a new proposal is presented below. For high-Reynolds-number flows it may be more appropriate to include nonlinear effects or effects due to the turbulence in the flow to find a local solution for the critical layer. No such applications have been made to this field.
D. H. Peregrine
84
For a realistic class of flows satisfying V(0)< 0,
v”(z) 2 0,
V ( z ) finite,
(4.30)
in - h 5 z I 0, Yih (1972) shows that if there is a critical layer at z = zlr then c must be real and the boundary conditions are not consistent with a solution for which w ( z l ) is nonzero. He incorrectly excludes the case w ( z l ) = 0. By multiplying Eq. (4.8) by the complex conjugate function w*(z) and integrating from - h to zl, he finds
after putting w ( z l ) = w( - h) = 0. Since conditions (4.30)imply U”/(U - c) is positive in - h I z -= zl, the only possible value for W ( Z ) in that interval is zero. There may be a discontinuity in w‘(z) at a singular point, and thus a solution regular for z 2 z1 and zero for z I zi is possible.? Direct examination of the equations of motion shows that this type of solution satisfies them, and it will be called a “surface layer solution.” Such a solution is also possible for other flows; they do not need to satisfy conditions (4.30),so that other solutions may also be possible in some cases. Where more than one solution is possible, the relevant one in any circumstance might be determined by solving an initial value problem. It seems reasonable thdtoif the waves are generated by surface disturbances or by disturbances above the critical layer then the surface layer solution is appropriate; but if the wave generation is by a disturbance extending below the critical layer, other possible solutions may be expected to be relevant. It is desirable to ascertain when the surface layer dispersion relation differs significantly from the “conventional” solution. As usual, the only case that is simple to investigate analytically is the linear profile, for example, U(Z) = U,(1
+ z/h).
” .
(4.32)
t When this result was communicated to Professor Yih, he agreed that it is possible for the differential equation (4.8) to have a solution with a discontinuous w’ at z = zl, a possibility that had simply escaped his attention. He notes, however, that for k = 0 the solution regular at z=z,is w=u-c,
and this cannot possibly satisfy the Gee-surface condition. Thus for long waves Yih’s conclusion still stands. How large k2 has to be in order to have a solution with discontinuous w’ can be decided by following the development in Yih’s paper ‘(1972, pp. 214-216) for the case U”(zl)= 0. The condition U”(z,)= 0 can now be removed since we admit a solution with a discontinuous w‘, and an estimate of k for the longest possible waves may be made using the long-wave approximation given in Section IV,B,4.
Interaction of Water Waves and Currents
85
This has the “conventional ” dispersion relation
+
k coth kh = g/( Uo - c ) ~ Uo/h(Uo - c).
(4.33)
If there is a critical layer at a depth hl, hl
= (UO- C)h/UO
1
(4.34)
and the surface layer dispersion relation is
+
k coth khl = g/(Uo - c ) ~ Uo/h(Uo - c).
(4.35)
The two dispersion relations can only differ appreciably if khl is sufficiently small for tanh kh, to be noticeably less than one, say kh, < 2. The dispersion relation (4.35) can be rewritten in this particular case as khl coth khl = 1 + gh/(Uo - c ) U o ,
(4.36)
the right-hand side of which has a minimum value 1 + gh/Ui for 0 I c IU o. For khl to be less than 2, the left-hand side of (4.36) must also be less than 2. Thus the surface layer dispersion relation differs significantly from the conventional one only if gh/Ui < 1.
(4.37)
But this condition is appropriate for the case c = O , where in fact they agree, so in practice the condition is U i %. gh,
(4.38)
which implies that the shear must be quite large. The two dispersion relations are plotted, in two different ways in Fig. 16 for the case
U i = 4gh0 .
(4.39)
These results are of academic interest only since such high-speed flows develop finite-amplitude waves as instabilities of greater practical importance (see Section IV,D). For short waves, for which the depth of the flow is not significant, it is straightforward to show that the surface layer dispersion relation is only significantly different if uo
> 30,
(4.40)
where o is the frequency of the waves relative to the surface water, that is, o - k U o . Again, this is a strong shear.
D. H . Peregrine
86
1.2
-
1.0
t
wh/U(O)
A/h
I
-
-
0
4-=+= 1
3
wh/UlO)
FIG.16. Dispersion curves for the velocity profile U ( z )= 2(gh)'"( 1 is for the surface layer solution.
+ z/h).The dashed line
4. Approximate Solutions
Approximations may be made for long waves and for short waves. For long waves, kh 6 1 and an appropriate way to write Eq. (4.7) is [p'/( U
- c)']'
= k2p/(U - c),,
(4.41)
which may be integrated twice to give the integral equation p ( z )= A
+ B j' ( U , - c)' -h
dz,
+ k2
z2
(" j-h j-h
(u1
- c ) : p ( z l ) dz, dz, ,
- c)
(4.42) in which the abbreviation
is used, and A and B are constants to be determined by boundary conditions. It is now easy to find successive approximations to p ( z ) as a power series in k.
Interaction of Water Waves and Currents
87
For flow over a rigid bottom, B = 0, and, setting A = 1 without loss of generality,
+ O(k6h6),
(4.44)
with a dispersion relation
(4.45)
This result is given by Thompson (1949), but the first approximation 0
g
j
dzl/(Ul - c)’ = 1
(4.46)
-h
is better known from Burns’ (1953) paper. The same approach can be used for problems where the flow is uniform except for a thin layer, e.g., a boundary layer at the bottom or at the free surface, or to the surface layer solution when that layer is thin. For example, consider waves on still, deep water with a thin wind-driven boundary layer of thickness h. The pressure perturbation p(z)must vary as exp(kz) below the layer, and if U is effectively zero at z = - h, matching p and p‘ with solution (4.42) leads to c Z A= kB,
(4.47)
and to the approximate dispersion relation
Not one of the dispersion relations (4.45),(4.46),and (4.48) is easy to use or interpret. Perhaps the simplest is the first approximation to stationary
88
D. H.Peregrine
waves on a surface jet. This is obtained by putting o = 0 in (4.48) and leads to the result (4.49)
given by Peregrine and Smith (1975). The integral is proportional to the momentum flow in the surface jet. A thin sheet of momentum flow at the surface of a fluid acts rather like a negative surface tension. Compare Eq. (4.49) with g
+ ( T k * / p ) = kc’,
(4.50)
the deep-water dispersion relation, when c = 0 (but also see “surface shear waves” in Section IV,C). At the other extreme, when waves are short compared with the current variations, wave properties are determined by the flow close to the surface. One can either use a WKB approximation for W ( Z ) as Dalrymple (1973, Appendix 1) does, or expand systematically in inverse powers of ko = duo - c)’,
(4.51)
as is done by Peregrine and Smith [1975, Eq. (48)]. The first three terms of the dispersion relation are (4.52)
where a zero subscript indicates that the function is evaluated at the surface. Further approximations involve higher derivatives of U ( z ) that would be difficult to evaluate from measurements of a real flow. Some idea of the accuracy of these approximations may be obtained from Fig. 17 which shows the dispersion relation for stationary waves on a deep flow of the form U ( z )= Uoea‘,
with k plotted against U o, using appropriate dimensionless variables. A few bounds for c are available. Thompson (1949) proves Umin- (gh)”’ Ic I U,,,
+ (gh)”’,
(4.53)
and that when U is a monotonic and nondecreasing function of height above the bed, c I U,,,
+ (g/k)’’’.
(4.54)
Interaction of Water Waves and Currents
1
-
2
89
3
s/u:
FIG. 17. Dispersion relation for two-dimensional stationary waves on the velocity profile U ( z )= LI, exp az, together with approximations. S,, S,, and S, are successive short-wave approximationsand L,is the first long-wave approximation. (From Peregrine and Smith, 1975, Fig. 4.)
With similar conditions,
U' 2 0 and finite,
U 5 0,
(4.55)
Yih (1972) extends a result of Burns (1953) to prove that there is one solution with c IU ( - h), (4.56) and another with c 2 U(0). (4.57) It is worth noting that there is no simple equivalent of the linear longwave equations for irrotational flow:
au ay -++-=o, at ax
ay
au ax
-+hh-=OO. at
There is only the result (4.46) for the long-wave velocity.
(4.58)
90
D . H . Peregrine
5. Upstream Propagation Conditions (4.55)are such that many profiles V ( z )that may be chosen to represent stream flow would satisfy them. If U ( - h) is zero, condition (4.56) indicates that that solution corresponds to upstream propagation of waves, regardless of how large the surface or mean velocities may be. Benjamin (1962,p. 108)suggests that the solution (4.56)may not be physically realizable when the mean velocity is much greater than (gh)’/’. He argues that the relatively high velocities near the bed due to the wave in such a solution severely limit the amplitude of the wave if separation of the boundary layer is not to occur. Yih (1972)discusses this further, confirming the high perturbation velocities near the bed, and arguing against a conjecture af Benjamin’s that the maximum velocity of propagation upstream should be of the order -(gh)’/’ + 0. There are several facets to this problem. There is no doubt of the existence of the mathematical solution corresponding to (4.56)with conditions (4.55) for any value of 0. The conditions (4.55)certainly apply to a real flow if the Reynolds number is low enough for it to be laminar. However, in that case it is less realistic to omit viscosity in the analysis. If viscosity is included these waves do not occur, as is shown by a stability analysis [see Benjamin (1957) or Yih (1969,Sect. 9.9)for further results]. For a turbulent high-Reynolds-number flow, many model profiles U ( z ) would give either a nonzero velocity at the bottom or an infinite velocity gradient U’(- h) as in the frequently used one-seventh power profile (4.1).In the latter case, Lighthill (1953)shows there is no upstream propagation for
e
U > 1.0353(gh)’’’.
(4.59)
There is also the problem of generation and detection of such waves. For example, using the linear velocity profile (4.34)a rough calculation shows that if a low-frequency oscillating surface pressure is applied over an appropriate length of the surface, the two long-wave modes are generated with amplitudes inversely proportional to their phase velocities. This means that for a high-Froude-number flow, the controversial upstream propagating mode would have a substantially smaller amplitude than the waves propagating downstream. Its group velocity upstream would also be small, so that it would suffer appreciable damping, by neglected effects, before it got clear of the generating area. In summary, these particular upstream propagating waves appear to be a mathematical solution with little physical relevance. There is an exception, when the flow separates from the bed. This is discussed further in Sections IV,C and D.
Interaction of Water Waves and Currents
91
6. Group Velocity
In many applications, the most important wave parameter is the group velocity; for example, it is needed to find the stopping velocity in a hydraulic breakwater. Once again, the only simple case is the linear velocity profile U(Z) = u,
+ ZU, .
(4.60)
The group velocity cu is given by
in which B(kh) is the ratio cU/c for waves of the same wave number in still water of depth h. The denominator of the right-hand side of (4.61) cannot be less than f since the maximum value of (c - U,)Ub/g is 1, which it attains at k = 0. From Eq. (4.61) one may see that (c,, - U,)/(c - U , ) behaves rather like B(kh) but “skewed” in the direction one would expect from the underlying shear. Inspection of Fig. 16 shows that the surface layer solution has similar properties, except at c = 0, a point that may merit further attention. For most applications, numerical solutions need to be found; even when analytic solutions are found it may be more convenient to determine c, graphically or numerically, e.g., Taylor (1955) finds the stopping velocities of a sectionally linear profile graphically.
C. FINITE-AMPLITUDE WAVES One familiar method of finding finite-amplitude wave solutions becomes relatively inappropriate when the basic flow varies with depth. This is the method of expanding the free-surface boundary condition in a Taylor series about the mean level. If this approach is adopted, the mean flow V(z) must also be expanded in a Taylor series. While this may be sensible for a flow chosen for its mathematical convenience, such as a linear profile with constant vorticity, it is quite inappropriate if actual velocity measurements are used, since even second derivatives may be quite uncertain. A number of transformations of the equations of motion for steady flow enable this problem to be avoided, at least for steady periodic waves. A few transformations are now given, followed by some solutions.
D. H. Peregrim
92
1. Transformations of the Equations
For steady flow in two dimensions, the introduction of a stream function $ leads to the expression -V2$ for vorticity and to the equation
v’* =f (*)
(4.62)
to express the fact that for inviscid flows vorticity is constant along streamlines (Batchelor, 1967, Sect. 7.4). Transformations of coordinates that effectively replace z with II/ thus have two desirable properties. The free surface becomes a fixed boundary, I) = const, and the basic distribution of vorticity is explicitly stated. The most obvious transformation is the direct one, (4.63)
u(x, z ) = u(x, *),
a von Mises transformation. The continuity equation and Eq. (4.62)become
au + w a u - u-a w = 0, a+ a* aw aw a u -+ w-+u-=f(*), ax a* a* -
ax
(4.64)
-
(4.65)
respectively. These are two equations for the two components of the total velocity in a reference frame moving with the wave. Gouyon (1958) and Moiseev (1960) have used these equations for existence proofs in the case where the variation in the basic flow is small compared with the wave velocity, so that to a first approximation they are additive. A less direct approach is to use the height z of a streamline as an independent variable, that is, z = z(x, *). (4.66) The total velocity components are then u = l/z$
and
w = z,/z*,
(4.67)
where subscripts are used to denote partial derivatives. The vorticity equation (4.62) becomes z,,z$ - 224ZZ,Z$+ Z$$(l
+ z:)
= Z;f(*).
(4.68)
Dubreil-Jacotin (1934) uses this equation for an existence proof. Dalrymple (1973) gives a finite-difference approximation to Eq. (4.68) and presents sample results of finite-amplitude waves on a linear shear and on a oneseventh power profile. Benjamin (1962) takes this approach a step further in his derivation of a solitary-wave solution. He introduces a new height variable s, equal to the
Interaction of Water Waves and Currents
93
value of z in the undisturbed flow. Thus the undisturbed flow is given by
*
=
w,
(4.69)
where
U ( S )- c = dY/ds.
(4.70)
The vorticity equation now becomes
22,,2,2,
+ Z,,(l + zf)} + vl(s){zj - z,( 1 +
Zf)}
= 0.
(4.71)
An advantage of this equation is that U ( s )appears explicitly. Note that it is a nontrivial matter to find f ($) to substitute in Eqs. (4.62), (4.65), and (4.68) for most velocity profiles, since it is given by
f(*) = yT”(s),
*
=
W).
(4.72)
2. Solutions The earliest finite-amplitude wave solution is Gerstner’s (1802, in Lamb, 1932, Sect. 251) and it has a vorticity distribution. As Lamb shows, following Stokes, the uniform flow corresponding to that vorticity distribution is
U ( z )= -ce2kb,
(4.73)
where k(z - z,) = kb - ie2kb,
c2 = g / k ,
kz, = 4a2k2 - In ak.
The flow is in the opposite direction to the waves’ propagation and for the highest wave the vorticity is singular at the free surface. This solution is unlikely to be relevant to waves on real flows. As might be expected from the difficulty of finding solutions for infinitesimal waves for most velocity profiles, the only analytic solution corresponding to the Stokes wave for irrotational flow is for flow with uniform vorticity. Tsao (1959) gives a third-order approximation for arbitrary depth. The algebraic complexity of the solution is somewhat daunting, despite the fact that for uniform vorticity it is possible to introduce a velocity potential for the wave motion. For the linear profile U = bz,
(4.74)
it is relatively simple to show that for deep-water waves, the wave motion is given by a velocity potential acekz sin k(x - c t ) + $a2b(3+ S)e2k’ sin 2k(x - ct) + O(a3k3),
(4.75)
D. H. Peregrine
94
with a surface elevation [ = u cos k ( -~ C t )
+ $i2k(l + 2 s + is’) cos 2k(x - ct) + O(a2), (4.76)
in which kc2 + bc - = 0
(4.77)
S = b/ck = b/a.
(4.78)
The constant in Bernoulli’s equation is increased by
+
C2 = fa2b(ck tb),
(4.79)
and this may be interpreted as either a change of level of the free surface by C2/(g - be) or an additional uniform velocity - C2/c. Tsao (1959), Eq. (2.19), is not in agreement with the result (4.79). From the wave elevation (4.76) it is easily seen that the usual asymmetry between crest and trough increases as the shear increases for waves traveling in the + x direction. That is, if the maximum current is at the surface, waves traveling in that direction have sharper crests than corresponding irrotational waves. Conversely, waves traveling in the opposite direction are more nearly symmetrical about the mean level. The wave profile is sinusoidal to second order when
s=-2+Jz.
(4.80)
s 2 -1
(4.81)
We may see that by rewriting the dispersion relation (4.77) in terms of S, c2(1
+ S) = g/k.
(4.82)
However, for c negative a surface layer solution should be found, which may modify the results. Numerical techniques to solve the finite-amplitude problem for steady waves have been developed by Dalrymple (1973, 1974). He extends Dean’s stream function method (Dean, 1965), which essentially is a double Fourier expansion of the stream function. A considerable number of results are presented for waves of flows with linear velocity profiles. The results include large-amplitude waves. Dalrymple (1973) also presents some results from a finite-difference approximation to Eq. (4.68) for other velocity profiles. These methods appear to be an effective approach to solving specific problems, and by comparison with irrotational waves, the effects of vorticity may be better understood.
Interaction of Water Waves and Currents
95
Solitary-wave solutions may be found for any velocity profile without a critical layer. The solution is given by Benjamin (1962) and has been derived by several other authors since then. The corresponding Korteweg de Vries equation is given by Benney (1966) [which appears to have an error in Eq. (54)] and by Freeman and Johnson (1970). The equation is (4.83) where (4.84) (4.85) and co is the linear long-wave velocity given by g l , = 1.
(4.86)
Note that the I, are negative for n odd. The solitary-wave solution is = a sech’ P(x
- c1 t),
(4.87)
in which c1 = co - aI,g/213,
(4.88)
8’
(4.89)
= a14g/4J.
These reduce to the usual irrotational results for water at rest far from the wave, c1 = (gh)”’( 1
+ $a/h),
P’
(4.90) Benjamin (1962) discusses the results for waves propagating on a stream and deduces that the effect of the vorticity is small unless the Froude number of the flow is near one. In that case the waves most commonly met are stationary waves. Strictly in such a case there is a critical layer at the bottom of the flow, but if the one-seventh power velocity distribution (4.1) is used to model the flow the analysis is not affected since all the integrals converge. A comparison, of stationary waves on such a flow, with the corresponding irrotational flow, is given in Fig. 18. The example calculated corresponds to the waves generated by a small obstacle, stationary in a stream with Froude number
F = U(gh)-”’.
= 3a/4h2.
(4.91)
96
D. H. Peregrine
FIG. 18. Stationary waves on a stream caused by a reduction pU2hD2in the momentum flow.(a) Uniform stream of velocity 0.(b) Stream with velocity profile. (8/7)0(z + h)'"h-"'.
It is supposed that the momentum flow (4.92)
is reduced by a small amount p02hD2,
(4.93)
Interaction of Water Waves and Currents
97
which corresponds to the force on the obstacle. The energy of the flow is assumed to be unchanged. [See Benjamin and Lighthill (1954) for a full discussion in the case of uniform flows, and Fenton (1973) for an application to a linear profile.] The right-hand boundary in the figures dorresponds to the solitary-wave solution. At first sight the slope of this curve is surprising, since the larger the wave the faster it travels. However, one result of the loss of momentum flow is to reduce the mean level of the stream, so there is no inconsistency in a larger wave being on a slower stream. The most noticeable difference between Figs. 18a and b is the increase with vorticity of the area of the (a,F 2 ) plane in which waves may occur. Essentially this is due to the increased speed of the solitary wave, which is noted by Benjamin (1962). Other differences are quantitative rather than qualitative. The biggest of these is an increase in wavelength for the stream with vorticity when F 2 c 0.85, but the theory is less appropriate there. For finite-amplitude shallow-water waves, the position is similar to that for linear long waves. Benney (1974) shows that there is no pair of “simple shallow water equations to characterize long waves in a general flow.” On the other hand, if attention is focused on waves propagating in one direction only, some progress has been made by Blythe et al. (1972). By looking for an equation of the form
( a r p t ) + c aypx = 0,
(4.94)
r,
where c is a function of equations leading to a simple-wave solution are found for flows without a critical layer. An example is given for the case where the flow has uniform vorticity, but no guidance is given for other less simple flows. A rather specialized class of finite-amplitude waves has been described by Peregrine (1974) and named “surface shear waves.” The basic flow configuration is a sheet of rapidly moving water traveling over water at rest or nearly so. Stationary waves may form on such a flow. Below weirs or sluices they may have an amplitude much greater than the initial thickness of the surface sheet. The wide range of conditions in which this form of wave occurs is shown by Moore and Morgan (1959), who call it a “wave hydraulic jump.” A very simple theory is possible when the Froude number
(Mhh2)”*
(4.95)
is large. Here h is the thickness of the jet and M its momentum flow,
5
0
p U 2 ( z ) dz. -h
(4.96)
98
D. H. Peregrine
To a first approximation the surface flow is deflected only by the pressure difference across it. That is the difference between atmospheric pressure and the approximately hydrostatic pressure in the slow-moving water beneath. The appropriate equation for the surface elevation is Mlc = pgr,
(4.97)
where IC is the curvature of the surface. This is also the “elastica” equation for the bending of a thin sheet of elastic material. Thus the shape of the waves is easily reproduced by bending a sheet of paper. The steepness of the waves is not limited in theory, and in practice it is easy to produce them with slopes greater than 30”, the maximum for Stokes waves. The crests and troughs are rounded and symmetrical. Peregrine (1974) also draws attention to superficially similar waves that occur on beaches in the backwash from surf, described in more detail in the next section. If their structure is also similar, then they are formed by the high-velocity backwash separating from the beach and riding up over a separation bubble. This hypothesis is supported by experiments with the wave hydraulic jump, where by raising backwater levels, it can be formed by separation of flow from the plane spillway of a weir. 3. Highest Waves For irrotational waves, Stokes (Lamb, 1932, Sect. 250) showed that the highest waves in steady motion have a 120” corner at their crest. The same method of local analysis can be applied to waves on a rotational flow, and Miche (1944, pp. 386-406) shows that the result is unchanged. Miche proceeds further and shows that vorticity affects the curvature each side of the crest. In particular, vorticity b gives a free surface
!(y2
6 = & !!+ 3 3 9
+ ...,
(4.98)
where 6 is measured from the downward vertical. Grant (1973) shows that for irrotational waves the next term in an expansion like (4.98) has a (probably) transcendental power of r approximately r 1.2. Delachenal(l973) also considers this problem but assumes that the vorticity has the form Ar- l‘zf (e),
(4.99)
which is singular at the crest. Thus his solution is in the same unrealistic class as Gerstner’s highest wave. A different aspect of highest waves is the maximum amplitude that may be attained. If the phase velocity of the wave in question is known, then the use
Interaction of Water Waves and Currents
99
of Bernoulli's theorem in a frame of reference moving with the wave gives the maximum amplitude corresponding to a stagnation point at the crest. Banner and Phillips (1974) draw attention to the effect a surface drift due to wind has in this context. If the surface drift velocity has magnitude q, then the maximum amplitude of a wave of phase velocity c relative to deep water in the same direction as q is (4.100) Note that for an irrotational wave of given wavelength c only varies by 20 % as the amplitude increases to its maximum. It is unlikely that a thin shear layer can cause a much larger variation. Thus if q/c is near 1, the maximum wave can be expected to have quite a small amplitude, and any small amplitude approximation is likely to be very limited in its applicability. As Banner and Phillips point out, this is likely to be the case for the shorter waves in a wind-driven sea. Phillips and Banner (1974) investigate the effect of large waves on a surface drift layer. The water motion in the large wave causes the surface drift to vary, with its maximum velocity at the crests of the long wave. As discussed in Section II,F, short waves become shorter near long-wave crests, and thus their phase velocity relative to the water below the surface layer decreases, just when the velocity of that layer increases. When q = c they cannot propagate, and however small their amplitude, it seems that they must break. Phillips and Banner estimate these effects using linear wave theory. If the long wave has appreciable steepness, then the amplification of the surface drift at the crest of the wave is substantial. Thus the interaction of long surface waves with a surface drift layer may have appreciable effects in suppressing shorter waves. The proportionate reduction in wave energy is estimated for the shorter waves and is found to be in surprisingly good agreement with experimental measurements. For irrotational waves, Longuet-Higginsand Fenton (1974) and LonguetHiggins (1975) show that the highest wave is not the wave with most mass, momentum, or energy. This is very relevant to both wave breaking and to actually producing a highest wave. Presumably, similar results are likely to hold for rotational waves.
D. STABILITY Except for flows in thin films, which are not considered in this work, the currents of interest are turbulent flows, so discussion of stability may seem inappropriate. This is not so. If an instability transfers energy to surface
100
D . H . Peregrine
wave motion, then it is of direct interest in the present context. Also it has been suggested, with respect to other turbulent flows, that “instabilities of the mean velocity may determine the large-scale structure of the flow (Landahl, 1967). The stability of inviscid flows (U(z),0, 0) over a rigid bottom to infinitesimal disturbances is considered by Yih (1972). He extends several theorems for flow between two rigid walls to this case and shows that the requirements for stability are very similar in that unstable modes are associated with inflection points in the velocity profile. Silcock (1975) examines the surface jet flows ”
U = sech z
and
U = exp( - j z 2 )
(4.101)
in detail and computes the growth rates of infinitesimal disturbances. The stationary-wave solutions form part of the stability boundary, but for small Froude numbers (based on the jet “thickness”) the growth rates of the associated instabilities are very small. There are also instabilities that have little effect on the surface. Unlike many topics in this paper, there are some experimental results. Sarpkaya (1957) reports on a substantial experimental work in which waves were propagated against a stream flowing under gravity. Measurements were made of wave phase velocity, amplitude, wavelength, and shape, for those particular waves that propagated unchanged in amplitude. Higher and shorter waves were amplified, and smaller and longer waves were damped. Figures 19 and 20 are taken from Sarpkaya (1957) and summarize some of the results. It may be noted that only waves of finite amplitude are amplified. It is odd that a set of neutral waves, varying in frequency, was not found for each flow.
0
1.o
0.5
1.5
value of R X ~ O - ~
FIG. 19. Stability boundaries of stream flows of diNerent Froude numbers with waves propagating upstream. Amplitude/wavelength is plotted against Reynolds number. (From Sarpkaya, 1957, Fig. 2, p. 575.)
Interaction of Water Waves and Currents 0.4(
I
I
I
101
I
F e 0.10 0.15 B 0.20 e 0.25 e 0.30 o 0.35
-
0.30 G \
10
1 c
0
-0.15
E
p0.2
0.20
L
-%
5
0,
e -(
0.10
I
I
I
!5
0.35 1.5
v m of ~
~ 1 0 - 5
FIG.20. Stability boundaries of stream flows of different Froude numbers with waves propagating upstream. Depth/wavelength is plotted against Reynolds number. (From Sarpkaya, 1957, Fig. 3, p. 575.)
It is of particular interest that there is no critical layer for any of the waves measured in these experiments; thus an explanation may need to include the interaction of the water waves and the turbulence in the flow, or interaction with the boundary layer on the bed of the channel. The latter seems most likely. By using the results shown and linear irrotational theory it is possible to work out u( - h)/O (that is, the particle velocity due to the waves at the bed divided by the mean velocity). For a high proportion of the experimental results, this ratio lies in the range 0.6-0.7with no systematic variation apparent. This may we11 be sufficient to cause separation or substantial thickening of the bottom boundary layer. A more marked instability occurs in uniform streams at high Froude numbers. Large-amplitude waves form and develop bores at their fronts. These progress downstream with variable frequency and amplitude. They are called roll waves. This instability may be demonstrated theoretically (Jeffreys, 1925) by using the linear long-wave equations for irrotational flow and adding a Chezy friction term, that is, a quadratic resistance term that also varies inversely with depth of water. Dressler (1949) gives more details of solutions and Dressler and Pohle (1953) consider more general friction
D. H . Peregrine
102
laws. Experimental measurements of the development of roll waves are given by Brock (1969), and numerical examples are calculated and discussed by Jolly and Yevjevich (1974). Friction laws for streams are one way of representing the turbulence that also gives rise to the mean velocity profile. Laminar flow down a plane is also unstable if the Reynolds number is greater than
2 cot p,
(4.102)
where is the inclination of the plane to the horizontal, and Benjamin (1957) points out the analogy with roll waves. More details may be found in Yih (1969, Ch. 9, Sect. 9) and experimental results in Benjamin (1961). Corresponding calculations for turbulent stream flow would require an eddy viscosity or other hypothesis to represent the turbulent Reynolds stresses. Another type of instability gives rise to the surface shear wave in backwash on a beach, mentioned at the end of Section IV,C,3 (Peregrine, 1974). When the backwash is not affected by a following wave it usually forms a nearly stationary turbulent bore where it meets the still water. After a time a long smooth wave may emerge in front of the bore and travel upstream, gaining height, usually until it dwarfs the bore it sprang from. If a hypothesis of a separation of the flow from the bed is correct, the wave may start in the following way. According to linear theory, a small disturbance, exponentially decaying upstream, may precede the bore. If this disturbance is sufficient to cause flow separation on the bed, the wave may appear. It will grow by entrainment of water into the separation “bubble.” It seems possible that such an instability may only occur for certain velocity profiles, e.g., flow that starts from rest on a slope, or may depend on the Reynolds number of the flow. E. WAVESON FLOWIN
CHANNELS
This pertains to waves on a flow
u = ( q y , z), 0, O),
(4.103)
confined in a channel. Peters (1966) treats the case of long waves, deriving the equation
jjdY d z / ( W , z ) -
C l 2 = h/g
(4.104)
S
for the long-wave velocity, in which S is the cross-sectional area of the channel and b its surface breadth.
Interaction of Water Waves and Currents
103
The linearized equation corresponding to Eq. (4.7) is (4.105)
with
dppn =o
(4.106)
on the walls of the channel and (4.107)
on the mean free surface. No solutions have been found with any y variation. Peters (1966) also finds the equation for a solitary-wave solution, but a subsidiary partial differential equation, similar to (4.107), must be solved in S . However long the waves are, such solutions are only likely to be of value for channels that do not have a large aspect ratio (width/depth). In the irrotational case, Peregrine (1968) shows that a second long-wave approximation (which is needed for the solitary-wave solution) has a term that increases with the square of the aspect ratio for nonrectangular channels. V. Turbulence
In considering the interaction of waves and turbulence, a useful way to develop ideas is to take a simple view of the turbulence. That is, characterize the turbulence by a length scale and a typical maximum fluctuation of velocity. Although turbulence has important small-scale properties, it seems likely that interactions are dominated by the most prominent turbulent motions, and the ratio of their lengths and velocities to those of the waves. Perhaps the simplest case to understand, though a difficult one to analyze, is when the turbulence has a scale much greater than the waves and velocities comparable with the wave group velocity. The waves are then refracted in accord with the equations derived in Section I1,C. However, consideration of the solutions in Sections II,D and E shows that unless the waves are reflected, by refraction, out of the region of turbulence, they are likely eventually to encounter currents causing their wavelength to diminish considerably so that a high proportion of their energy is lost by breaking. Thus large-scale turbulence acts as a wave absorber. Such behavior is easy to see on rivers, where wind waves get little chance to grow if the turbulence is strong enough. Usually this seems to coincide with a level of turbulence, which noticeably deforms the free surface so that its dominant features are small ripples and dips above vortex cores.
104
D . H.Peregrine
Much stronger turbulence leads to relatively violent surface motions that can generate some propagating waves, or in extremes as in turbulent hydraulic jumps, it leads to the surface breaking up into drops and irregular masses of water. The whole range of behavior may be observed, on a relatively small scale, in the boundary layer of ships. An example that is commonly observed is the effect of a ship’s wake on short wind waves that are incident upon it. The turbulence is relatively strong and often of larger scale and thus acts to absorb or reflect the waves by refracting and steepening them. However, the mean motions associated with a wake are also likely to be important. There is the flow along the wake in the direction of motion of the generating vessel and also the transverse motions due to trailing vortices (from bilges or propellors or both). That these latter may be dominant is indicated by the relatively stronger effect that a curved wake has on waves. When the turbulent velocity fluctuations of large-scale turbulence are weak compared with the wave velocity, one may think in terms of waves being scattered by the turbulence. Phillips (1959) attempts to analyze this scattering for very weak turbulence, using a Fourier decomposition of the velocity field. It is difficult to make use of such an approach since the components of a spatial Fourier decomposition of turbulence are virtually unknown except for very special cases. Phillips uses estimates based on the inertial subrange of turbulence, but the author does not think this is likely to be an important part of the interaction. It seems somewhat more likely that Howe’s (1973) method for treating scattering may be applicable. Turning to the other extreme, we have small-scale turbulence. This may actually be the same turbulent flow but viewed with respect to different, longer waves. Small-scale turbulence is more amenable to study in laboratory experiments and to the extension of empirical methods established in other fields. For example, the “ friction laws ” established for steady flows in channels are often extended to unsteady flows such as long waves (see the case of roll waves mentioned in Section IV,D). On the other hand, Sarpkaya’s (1957) experiments (also mentioned in Section IV,D) show that some waves on a turbulent flow are amplified. Taking another viewpoint, the rate of strain tensor for a plane irrotational wave train is eij = a24/axi axj .
(5.1) This is oscillatory, but as Phillips (1959) points out, the Stokes’drift may be more important. More precisely, the finite-strain tensor has a component of second order in ak increasing linearly with time. It acts to stretch vortex lines and thus may lead to stronger interactions than the oscillatory part. Experiments by Green et al. (1972), described below, appear to give some support
Interaction of Water Waves and Currents
105
to this idea. If it is important, then the common eddy viscosity hypothesis may be of limited value. A few experiments have been performed to measure the scattering and dissipation of water waves by turbulence. In none of these experiments has scattering attributable to the turbulence been detected. The most interesting, from the viewpoint of the rest of this paper, are two experiments Savitsky (1970) reports. In both experiments turbulence was generated by towing a grid in a ship model testing tank. Waves were sent along the tank to overtake the grid and its turbulence. In the first experiment the grid spanned a 12-ft (3.7 m) tank. In the second experiment, the grid was only 3 ft (0.9 m) wide in a 75-ft (23 m) tank. Savitsky was unable to detect any scattering or dissipation by the turbulence, since in both cases the effects of the mean currents set up by the moving grid dominated the wave behavior. In the first experiment, there was a velocity defect at each side of the tank and the waves became unsteady with curving crests. In the second experiment the waketype flow refracted and diffracted the waves. The maximum mean velocities in these experiments were more than 10%of the group velocity of the waves. A more successful experiment is reported by Green et al. (1972). In a laboratory tank, turbulence was generated by a grid oscillating vertically. A false bottom was inserted, for all but the longest waves, to protect the surface layer of water from mean currents. The turbulent eddies had a scale of around 1 cm and the shortest waves had a wavelength of about 5 cm. Damping of the waves was observed. Measurements were made on various waves before and after propagating through the turbulent region, the turbulence being approximately the same in all cases. The results are presented in two different ways, one assuming an exponential decay with distance, the other assuming a quadratic decay law daldx = - ya2 (5.2) for the turbulent damping. The coefficients obtained from the measurements show appreciable scatter, but assumption (5.2) gives the smaller scatter. The coefficient y is found to depend on frequency in such a way that y = Cb5,
(5.3)
where C is a dimensional constant. Green et al. (1972) note Phillips’ (1959) suggestion that the most effectiveinteraction with the turbulence may be of second order in ak. Also, one may note that the time that any wave group spends in the turbulent region is inversely proportional to its group velocity. For this purpose it is adequate to assume the linear deep-water gravity wave dispersion relation, in which case Eqs. (5.2) and (5.3) may be rewritten daldx = - Cg3a2k2/2c,, (5.4) which supports the hypothesis that second-order effects are relevant.
106
D. H . Peregrine
A rather different experiment is reported by Green and Kang (1976). In this case, a single long wave, the fundamental resonant mode of a wave tank, was allowed to decay in the presence of different intensities of turbulence. The turbulence was due to thermal convection generated by heating the bottom of the tank. The intensity of this turbulence depends on the Rayleigh number. A major problem in interpreting the results of this experiment is that less than 20% of the observed damping could be ascribed to turbulence. However, after careful analysis and estimation of the other dissipative effects, Green and Kang present results for the turbulent damping that show an appreciable dependence on Rayleigh number. They also provide an interpretation of the interaction. Convective turbulence takes the form of intermittent thermals arising from the bottom boundary layer. When there is a horizontal flow, such as that due to long waves, a rising thermal carries relatively stationary fluid from the bottom boundary layer into the main moving mass of fluid. This can be interpreted as giving a Reynolds stress approximately equal to npuw, where n is the fraction of the horizontal area over which thermals occur at any instant, u the horizontal velocity of the main mass of fluid, and w a typical vertical velocity in a thermal. Green and Kang’s (1976) results are consistent with the estimate of this Reynolds stress that they make.
VI. Ship Waves A major aim in the study of ship hydrodynamics is the prediction of the total hydrodynamic resistance of a ship. This is a difficult and complex problem, so that another more practical topic is also studied: how to relate measurements on ship models to the behavior of the prototype. The traditional approach has been to divide the resistance into two parts, “viscous resistance” and “wave resistance,” and to scale the first according to the Reynolds numbers of the ship and its model and scale the second with their Froude numbers. A similar approach is used for studying the primary problem of predicting ship resistance theoretically. Independent methods of measuring the viscous and wave components of resistance, by measuring the water velocities and wave amplitudes behind ship models, have shown that this simple view is inadequate. Explanation of the actual resistance involves consideration of the interaction between waves generated by a ship and the flow around and in the wake of the ship. Important papers illustrating this point are those by Lackenby (1965) and Shearer and Cross (1965). The wave resistance of ships is the subject of a recent substantial survey by Wehausen (1973), which gives more details on many of the topics mentioned here.
Interaction of Water Waves and Currents
107
There are different ways of looking at this subject. One can look at physical quantities and interpret them directly, or, from a theoretical viewpoint, start with some approximation and interpret higher-order terms as interactions. For example, a direct physical approach to a finite-amplitude water wave does not lead to the concepts of linear and nonlinear waves that arises from mathematical approximations. However, interpreting and estimating the physical quantities depends on adequate theoretical backing. A “physical” analysis of ship resistance is illustrated in Fig. 21, which is an adaptation and extension of a diagram in Brard (1972, Fig. 2). Some of the subdivisions in the diagram are not easy to define, especially with respect to form drag, but it does help to understand the exchanges between the viscous and wave components. Two different ways these may be defined are indicated by the dotted lines. However, in neither case can the viscous component be expected to be entirely independent of the Froude number or the wave component independent of the Reynolds number. The direct viscous drag is approximately dependent on the wetted surface area, but this depends both on the shape of the water surface around the ship and on the trim of the ship. Both of these depend on the waves generated by the ship in its own vicinity. In the same way the form drag, which may be largely due to regions of flow separation and to trailing vortices, depends on the same two “ wave ” variables, waterline and trim. When waves break, as is very often the case near the bows of ships, momentum is transferred from the wave motion into the water. Even for steady ship motion, breaking can be an unsteady process in which case there can be momentum transfer into other wave components. For full-size ships, breaking is probably the most important form of wave dissipation, but there is also some due to turbulence in the boundary layer and wake and due to viscosity. At the lower Reynolds number of ship models, viscosity may be more important and surface tension effects near wave crests can also be relevant. The current field associated with flow around a ship may also be considered a wave generator. It is propagating into still water at the same speed as the ship. This notion is given approximate quantitative form in the papers by Beck (1971), Brard (1972), and Tatinclaux (1970). There is some imprecision here since this effect could also be termed a wave-current interaction. Most wave-current interactions involve a transfer of momentum between the two components. The mean flow and waves are steady in a frame of reference moving with the ship, and as the work in Section I1 indicates, in a steady situation wave action flux is conserved in those cases where it can be defined; but conservation of wave action does not normally imply conservation of momentum in one part of the system. The waves generated by the ship interact with the flow around it, the approximately inviscid flow, as well as the boundary layer and wake.
D. H.Peregrine
108
G Total resistance
I/ 1
..
J
of s t r e s s over ship's surface
tangential s t r e s s
Direct viscous drag
/normal
Form drag
stress
Direct wave drag
Momentum flow through a control
a Total resistance
FIG.21. An analysis of the hydrodynamic resistance to a ship's motion. The quantities within ovals are often measured for ship models. The dotted lines indicate two different divisions into viscous and wave resistance.
Interaction of Water Waves and Currents
109
While the mathematical problem to be solved in predicting ship resistance can be stated, its solution requires a number of substantial approximations and simplifications. The usual approximations are to assume inviscid irrotational flow, a linearized free-surface boundary condition, and some geometrical ratio, such as breadth/length, to be small. Such approximations may be thought of as a first term in a perturbation expansion, but with so many parameters in the problem it needs careful analysis when looking for higher approximations to ensure a consistent approach. This is discussed in Wehamen (1973), so only the current-wave interactions are mentioned here. There are two particularly important aspects in the mathematical problem of the interaction of the waves generated by the ship and the flow around it. One is the interaction with the potential flow, and the other is the interaction with the boundary layer and wake. If the inviscid problem were solved without any approximation involving the speed and shape of the ship, then the first-named interaction would be automatically satisfied. However, this is not usually the case. (Note that unless the ship is very slender or deeply submerged, linearizing the freesurface boundary condition is a poor approximation in the vicinity of the ship.) The ship is usually taken to be “slender ” or “thin,” or else the Froude number is supposed to be small or large. The thin-ship approximation does take some wave interaction into account at first approximation; the sinkage and trim of the vessel may be calculated. However, the flow around (or perhaps one should say along) the ship and the wave motion are both small, so that interactions come in at the second approximation. On the other hand, if the ship is assumed to have finite bulk and a low Froude number, the first approximation has no waves, so the second approximation necessarily includes solving the wave pattern on the flow field of the first approximation. Dagan (1972) gives an interesting account of this last type of problem, using as a simple example a two-dimensional submerged body. Submerged bodies introduce some further considerations (e.g., see also Farell and Guven, 1973)but two-dimensional potential flow is much simpler than flow in three dimensions. The interaction of waves with the turbulent flow in the boundary layer and wake is also a difficult problem, even when only mean flows are considered. The approach of representing both the boundary layer and wake by a corresponding displacement thickness, that is, taking a semiinfinite body that is an appropriate amount larger than the ship, has been tried several times with only a relatively small improvement in the results. However, other approximations made at the same time may be more important. Another approach is to assume that the flow is inviscid but has a vorticity distribution that is chosen to model the actual flow. This seems better than introducing a simple eddy viscosity since the effect of the eddy viscosity on
110
D. H.Peregrine
the waves may not be representative of the effect of the effect of the turbulence (see Section V). The simplest example is to consider a wave-making source at the center of a “wake ” that is uniform in the direction of motion of the source. Peregrine (1971) uses this model and simplifies the analysis by supposing the waves are so short that the “ray theory” of Section I1 applies. The resulting wave pattern differs from that for a point source in motion through water at rest. The envelope of wave cusps is inside the Kelvin angle of 1%”once the maximum wake velocity is greater than 0.1 times the velocity of the wave-making source; the transverse waves behind the source are strongly distorted, although since these are the longest waves the approximation is less likely to be accurate. No information is given about wave amplitudes and with any ray theory approach to the problem initial values on a ray for the wave amplitude are difficult to ascertain. However, the approach is easy to understand. Rather more detailed descriptions of wake flow are used by Tatinclaux (1970) and Beck (1971). The former has a distribution of vorticity to represent the wake behind a thin vertical two-dimensionalcylinder of ogival cross section. Beck uses vortex sheets to model the wake behind a thin ship. Both papers assume that fluid velocities are small and use linearized boundary conditions at the free surface. The contribution of the wake to the wave resistance is calculated for one ship in each paper. Tatinclaux (1970) chooses a particular vorticity distribution, which decays relatively rapidly behind the cylinder, and calculates solutions for a range of Froude numbers. The wake has most effect for Froude numbers less than 0.5. It increases the wave resistance by an amount that varies considerably with Froude number, in an oscillatory manner, from over + 10 to - 35 %. Beck (1971) considers variation of the dimensions of the wake in his model. The effect of his wake is around f 10% of the irrotational wave resistance. All these papers indicate the importance of wave-wake interaction. A very direct experiment on the interaction of waves and a wake has been performed by Gadd (1975). Two identical ship models were towed in a catamaran arrangement. Where the bow waves of the models intersected, a steep pyramidal wave formed at high enough speeds. A vertical flat plate was introduced along the centerline between the two hulls, so that its trailing edge was just ahead of the steep pyramidal wave. This meant that the bow waves of the twin hulls met the wake of the plate. The introduction of the plate considerably modified the steep wave. It flattened and moved forward the wave peak and caused extensive turbulent flow. A large superficially similar wave often occurs with its crest at a ship’s stern. This experiment is expected to give an insight into the interaction between that wave and the ship’s boundary layer and wake. The behavior of this flow brings to mind Banner and Phillips’ (1974)
Interaction of Water Waves and Currents
111
paper, which is discussed in Section IV,C. The effect of the reduced velocity in the boundary layer relative to the waves is a decrease in the maximum height attainable. For example, if the velocity of the flow is one-half the ship’s speed, use of Eq. (4.100) shows that the maximum elevation of the water is one-quarter its maximum for irrotational flow. Gadd’s (1975, Fig. 3) photograph shows that the wave is breaking. One consequence of this observation is that one must expect nonlinear effects to become important at much lower amplitudes in this type of problem than in cases where irrotational flow is a good approximation. Further details, including wake traverses and wave measurements, are included in Gadd’s (1975) paper. They show more details of the interaction that occurs in this experiment. Interesting points are the appreciable changes in the wake behind the hulls when the plate is introduced and the associated differences in the waterline near the stern of the models and in the waves radiated. These appear to be largely due to the bow wave of one hull influencing the stern of the other and its modification when the plate is introduced. ACKNOWLEDGMENTS I wish to thank the many people who have assisted by sending me preprints, reprints, and copies of reports and inaccessible papers. The report by Dalrymple (1973) was particularly useful for Section IV,C. REFERENCES
ARTHUR,R. S. (1950). Refraction of shallow water waves: the combined effect of currents and underwater topography. Trans. Amer. Geophys. Union 31, 549-552. BANNER,M. L, and PHILLIPS 0. M. (1974).On the incipient breaking ofsmall scale waves. J . Fluid Mech. 65, 647-656. BARBER, N. F. (1949).The behavior of waves on tidal streams. Proc. Roy. SOC.,Ser. A 198,81-93. BATCHELOR, G . K. (1967). “An Introduction to Fluid Mechanics.” Cambridge Univ. Press, London and New York. BECK,R. F. (1971).The wave resistance of a thin ship with a rotational wake. J. Ship Res. 15, 196-216. BENJAMIN, T. B. (1957).Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554-574. BENJAMIN, T. B. (1961).The development of threz-dimensional disturbances in an unstable film of liquid flowing down an inclined plane. J . Fluid Mech. 10, 401-419. BENJAMIN, T. B. (1962).The solitary wave on a stream with an arbitrary distribution of vorticity. J. Fluid Mech. 12, 97-116. BENJAMIN, T. B. (1967).Instability of periodic wavetrains in nonlinear dispersive systems. Proc. Roy. SOC., Ser. A 299, 59-75. BENJAMIN, T. B., and LIGHTHILL, M. J. (1954).On cnoidal waves and bores. Proc. Roy. SOC.,Ser. A 144, 425-451.
D. H.Peregrine
112
BENNEY, D. J. (1966). Long nonlinear waves in fluid flows. J. Math. Phys. (Cambridge, Mass.) 45, 52-63.
BENNEY, D. J. (1974). Interaction and dispersion of long waves. Stud. Appl. Math. 53, 137-144. BETTS,P. L. (1970). Waves on a stream of finite depth which has a velocity defect near the free surface. J. Fluid Mech. 41, 509-521. BLYTHE,P. A., KAZAKIA, Y.,and VARLEY, E. (1972). The interaction oflarge amplitude shallowwater waves with an ambient shear flow: non-critical flows. J . Fluid Mech. 56, 241-255. BRARD,R. (1972). The representation of a givenship form by singularity distributions when the boundary condition on the free surface is linearized. J . Ship Res. 16, 79-92. BRETHERTON, F.P. (1971). The general linearized theory of wave propagation. In “Mathematical Problems in the Geophysical Sciences’’ (W. H. Reid, ed.), Lectures in Applied Mathematics, No. 13, Vol. 1, pp. 61-102. Amer. Math. SOC.,Washington, D.C. C. J. R. (1968). Wavetrains in inhomogeneous moving media. BRETHERTON, F. P., and GARRETT, Proc. Roy. SOC.,Ser. A 302, 529-554. BROCK,R. R. (1969). Development of roll-wave trains in open channels. J . Hydraul. Din, Proc. Amer. SOC.Civil Eng. 95, 1401-1427. BULSON,P. S. (1963). Large scale bubble breakwater experiments. Dock Harbour Auth. 44, 191-197.
BURNS,J. C. (1953). Long waves in running water (with appendix by M. J. Lighthill). Proc. Cambridge Phil. SOC.49, 695-706. CASEY,T. B. (1974). Freak waves off Ushant. Letter to the Editor. Lloyds List Sept. 6th, p. 2. CHU,V. H., and MEI,C. C. (1970). On slowly-varying Stokes waves. J . Fluid Mech. 41,873-887. CHU, V. H., and MEI,C. C. (1971). The non-linear evolution of Stokes’ waves in deep water. J . Fluid Mech. 47, 337-351. CRAIK,A. D. D. (1968). Resonant gravity-wave interactions in a shear flow. J. Fluid Mech. 34, 531-549.
CRAIK,A. D. D. (1970). A wave-interaction model for the generation of windrows. J . Fluid Mech. 41, 801-821. CRAPPER, G. D. (1957). An exact solution for progressive capillary waves ofarbitrary amplitude. J . Fluid Mech. 2, 532-540. CRAPPER,G. D. (1970). Non-linear capillary waves generated by steep gravity waves. J . Fluid Mech. 40,149-159. CRAPPER, G. D. (1972). Non-linear gravity waves on steady non-uniform currents. J . Fluid Mech. 52, 713-724. DAGAN,G. (1972). Non-linear ship wave theory. Symp. Naoal Hydrodyn. 9th Paris. Off. Naval Res., Arlington, Virginia. DAGAN,G. (1975). Taylor instability of a non-uniform free-surface flow. J. Fluid Mech. 67, 113-123.
DALRYMPLE, R. A. (1973). “Water Wave Models and Wave Forces with Shear Currents,”Tech. Rep. No. 20. Coastal Oceanogr. Eng. Lab., University of Florida, Gainesville, Florida. DALRYMPLE, R. A. (1974). A finite amplitude wave on a linear shear current. J. Geophys. Res. 79, 4498-4504.
DEAN,R. G. (1965). Stream function representation ofnon-linear ocean waves. J. Geophys. Res. 70, 4561-4572.
DELACHENAL, M. B. (1973). Existence d’ecoulement permanent de type coin pour un fluide parfait a surface libre. C. R . Acad. Sci., Ser. A 276, 1021-1024. DRAPER, L. (1957). Attenuation of sea waves with depth. Houille Blanche 12, 926-931. DRAZIN,P. G., and HOWARD, L. N. (1966). Hydrodynamic stability ofparallel flow of inviscid fluid. Adoan. Appl. Mech. 9, 1-89.
Interaction of Water Waves and Currents
113
DRESSLER, R. F..(1949). Mathematical solution of the problem of roll-waves in inclined open channels. Commun. Pure Appl. Math. 2, 149-194. DRESSLER, R. F., and POHLE, F. V. (1953). Resistance effects on hydraulic instability. Commun. Pure Appl. Math. 6, 93-96. DUBREIL-JACOTIN, M. L. (1934). Sur la dktermination regoureuse des ondes permanentes pkriodiques d’amplitude finie. J. Math. [9], 13(III), 217-291. EVANS,D. V. (1975).The transmission ofdeep water waves across a vortex sheet. J . Fluid Mech. 68, 389-401. EVANS,J. T. (1955). Pneumatic and similar breakwaters. Proc. Roy. SOC., Ser. A 231,457-466. FARELL, C., and GWEN,0. (1973). On the experimental determination of the resistance components of a submerged spheroid. J. Ship Res. 17, 72-79. FENTON,J. D. (1973). Some results for surface gravity waves on shear flows. J . Inst. Math. Its Appl. 12, 1-20. FRANCIS, J. R. D., and DUDGEON, C. R. (1967).An experimental study ofwind-generated waves on a water current. Quart. J. Roy. Meteorol. SOC.93, 247-253. FREDS~E, J. (1974).Rotational channel flow over small three-dimensional bottom irregularities. J. Fluid Mech. 66, 49-66. FRBEMAN, N. C., and JOHNSON, R. S. (1970). Shallow water waves on shear flows. J. Fluid Mech. 42, 401-409. GADD,G . E. (1975). An interaction between surface water waves and a turbulent boundary layer and wake. Trans. Roy. Inst. Naval Architects 117. GARGETT, A. E., and HUGHES,B. A. (1972). On the interaction of surface and internal waves. J. Fluid Mech. 52, 179-191. GOLDSTEIN, M. E. (1974). Aeroacoustics. N A S A Spec. Publ. SP-346. GOUYON,R. (1958). Sur les houles planes en profondeur infinie. C. R . Acad. Sci. 247, 33-35, 180-182. GRAHAM, E. W., and GRAHAM, B. B. (1969). Effect of a shear layer on plane waves of sound in a fluid. J. Acoust. Sac. Amer. 46, 169-175. GRANT,M. A. (1973).The singularity at the crest ofa finite amplitude progressive Stokes wave. J. Fluid Mech. 59, 257-262. GREEN,J. L. (1961). Pneumatic breakwaters to protect dredges. J. Waterways Harbors Din, Proc. Amer. SOC. Civil Eng. 87(May), 67-87. [Discuss., 87(Nov.), 127-136; 88(Aug.), 171-174 (1962).] GREEN,T., and KANG,S. W. (1975). Long-wave decay due to convective turbulence. J. Fluid Mech. 5 427-444. GREEN, T., MEDWIN, H, and PAQUIN,J. E. (1972). Measurements of surface wave decay due to underwater turbulence. Nature (London), Phys. Sci. 237, 115-117. HALES,L. Z., and HERBICH, J. B. (1972). Tidal inlet current-ocean wave interaction. Proc. Coast Eng. Conf:13th 1,669-688. HASSELMANN, K. (1971).On the mass and momentum transfer between short gravity waves and larger-scale motions. J. Fluid Mech. 50, 189-205. HAYES,W. D. (1970).Conservation of action and model wave action. Proc. Roy. SOC., Ser. A 320, 187-208, HAY- W. D. (1973). Group velocity and nonlinear dispersive wave propagation. Proc. Roy. Soc., Ser. A 332, 199-221. HOGBEN,N. (1974). “Fluid Loading on Offshore Structures. A State of Art Appraisal: Wave Loads,” Maritime Technol. Monogr. No. 1. Roy. Inst. Naval Architects, London. HOLLIDAY, D. (1973). Nonlinear gravity-capillary surface waves in a slowly varying current. J. Fluid Mech. 57, 797-802.
114
D. H.Peregrine
HOWE,M. S. (1973). On the kinetic theory of wave propagation in random media. Phil. Trans. Roy. SOC.London, Ser. A 274, 523-549. HUANG,N. E., CHEN,D. T., TUNG,C.-C., and SMITH,J. R. (1972). Interactions between steady non-uniform currents and gravity waves with applications for current measurements. J. Phys. Oceanogr. 2, 420-431. R. W. (1961). Interaction between gravity waves and a shear flow. HUGHES,B. A,, and STEWART, J. Fluid Mech. 10, 385-400. HYDROGRAPHIC DEPARTMENT (1973). “Tidal Stream Atlas: The English and Bristol Channels,” Publ. NP250.Hydrogr. Dep., Min. Defence, Taunton, England. JAMES,I. D. (1974). A non-linear theory of longshore currents. Estuarine Coastal Mar. Sci. 2, 235-249. JEFFREYS, H. J. (1925). The flow of water in an inclined channel of rectangular section. Phil. Mag. [6] 49, 793-807. JOHNSON,J. W. (1947). The refraction of surface waves by currents. Trans. Amer. Geophys. Union 28, 867-874. JOLLY,J. P, and YEVJEVICH, V. (1974). Stability of single-peaked waves. J. Hydraul. Res. 12, 315-355. JONES,D. S., and MORGAN, J. D. (1972). The instability of a vortex sheet on a subsonic stream under acoustic radiation. Proc. Cambridge Phil. SOC.72, 465-488. JONES,D. S., and MORGAN,J. D. (1974). A linear model of a finite-amplitude Helmholtz instability. Proc. Roy. Soc., Ser. A 338, 17-41. C., and WANG,J. D. (1970). Interaction between waves and JONSSON,I. G., SKOUGAARD, currents. Proc. Con& Coastal Eng., 12th 1, 489-508. KENYON,K. (1971). Wave refraction in ocean currents. Deep Sea Res. 18, 1023-1034. KING,R., PROSSER, M. J., and JOHNS,D. J. (1973). On vortex excitation of model piles in flowing water. J. Sound Vibration 29, 169-188. LACKENBY, H. (1965). An investigation into the nature and interdependence of the components of ship resistance (with appendices by E. F. Relf and J. P. Shearer). Trans. Inst. Naval Architects 107, 474-486. (Discuss., pp. 486-501.) LAMB,H. (1932). “Hydrodynamics.” Cambridge Univ. Press, London and New York. LANCZOS, C. (1961). “Linear Differential Operators.” Van Nostrand-Reinhold, Princeton, New Jersey. LANDAHL, M. T. (1967). A wave-guide model for turbulent shear flow. J. Fluid Mech. 29, 441-459. LE BLOND,P. H., and TANG,C. L. (1974). On energy coupling between waves and rip currents. J. Geophys. Res. 79, 811-816. LEWIS,J. E, LAKE, B. M., and KO, D. R. S. (1974). On the interaction of internal waves and surface gravity waves. J. Fluid Mech. 63, 773-800. LIGHTHILL,SIR JAMES(1972). The propagation of sound through moving fluids. J. Sound Vibration 24, 471-492. LIGHTHILL, M. J. (1953). On the critical Froude number for turbulent flow over a smooth bottom. Proc. Cambridge Phil. SOC.49, 704-706. LIGHTHILL, M. J. (1965). Contributions to the theory of waves in nonlinear dispersive systems. J. Inst. Math. Its Appl. 1, 269-306. LIGHTHILL, M. J. (1967). Some special cases treated by the Whitham theory. Proc. Roy. SOC., Ser. A 299, 28-53. (Discuss., J. E. Feir, pp. 54-58.) LONGUET-HIGGINS, M. S. (1953). Mass-transport in water waves. Phil. Trans. Roy. Soc. London, Ser. A 245, 535-581. LONGUET-HIGGINS, M. S . (1963). The generation of capillary waves by steep gravity waves. J. Fluid Mech. 16, 138-159.
Interaction of Water Waves and Currents
115
LONGUET-HIGGINS, M. S. (1969). A non-linear mechanism for the generation of sea waves. Proc. Roy. SOC.,Ser. A 311, 371-389. LONGUET-HIGGINS, M. S. (1972).Recent progress in the study of longshore currents. I n “Waves on Beaches and Resulting Sediment Transport (R. E. Meyer, ed.), pp. 203-248. Academic Press, New York. LONGUET-HIGGINS, M. S. (1975). Integral properties of periodic gravity waves of finite amplitude. Proc. Roy. SOC.,Ser. A 342, 157-174. LONGUET-HIGGINS, M. S., and FENTON, J. D. (1974). On the mass, momentum, energy and circulation of a solitary wave, 11. Proc. Roy. Soc., Ser. A 340, 471-493. LONGUET-HIGGINS, M. S., and STEWART, R. W. (1960). Changes in the form of short gravity waves on long waves and tidal currents. J. Fluid Mech. 8, 565-583. LONGUET-HIGGINS, M. S., and STEWART, R. W. (1961). The changes in amplitude of short gravity waves on steady non-uniform currents. J. Fluid Mech. 10, 529-549. LONGUET-HIGGINS, M. S., and STEWART, R. W. (1964). Radiation stress in water waves, a physical discussion with application. Deep Sea Res. 11, 529-562. LUDWIG,D. (1966).Uniform asymptotic expansions at a caustic. Commun. Pure Appl. Math. 19, 215-250. LUKE,J. C. (1967). A variational principle for fluid with a free surface. J. Fluid Mech. 27, 395-397. MCKEE,W. D. (1974). Waves on a shearing current: a uniformly valid asymptotic solution. Proc. Cambridge Phil. SOC.75, 295-301. MCKEE,W. D. (1975). A “two turning-point” problem in fluid mechanics. Math. Proc. Cambridge Phil. SOC.77, 581-590. MALLORY, J. K. (1974). “Abnormal Waves on the South East Coast of South Africa.” Univ. of Cape Town Libraries, Cape Town. MEI, C. C. (1973). A note on the averaged momentum balance in two dimensional water waves. J. Mar. Res. 31, 97-104. MICHE,M. (1944). Mouvements ondulatoires de la mer en profondeur constante ou decroissante. Ann. Ponts Chaussees 114,25-78, 131-164, 270-292, 386-406. MILES,J. W. (1957). On the reflection of sound at an interface of relative motion. J . Acoust. SOC. Amer. 29,226-228. MOISEEV, N. N. (1960). Theorem of the existence and uniqueness of rotational waves of periodic type. Prikl. Mat. Mekh. 24, 711-714. (Engl. trans]., J. Appl. Math. Mech. 24, 1058-1063.) MOORE,W. L., and MORGAN,C. W. (1959). Hydraulic Jump at an abrupt drop. Trans. Amer. SOC.Civil Eng. 124, 507-524. MUNK,W. H., MILLER,G. R., SNODGRASS, F. E, and BARBER, N. F. (1963).Directional recording of swell from distant storms. Phil. Trans. Roy. SOC.London, Ser. A 255, 505-584. NECE,R. E., RICHEY,E. P., and RAO,V. S. (1968). Dissipation of deep water waves by hydraulic breakwaters. Proc. Con$ Coast Eng., I I th 2, 1032-1048. NODA,E. K. (1974). Wave induced nearshore circulation. J. Geophys. Res. 79. 4097-4106. NODA,E. K., SONU,C. J., RUPERT,V. C., and COLLINS,J. I. (1974). “Nearshore Circulations under Sea Breeze Conditions and Wave-Current Interactions in the Surf Zone,” Tetra Tech. Rep. TC-149-4, Pasadena, California. PEARCEY, T. (1946).The structure of an electromagnetic field in the neighborhood of a cusp of a caustic. Phil. Mag. [7] 37, 311-317. PEREGRINE, D. H. (1968). Long waves in a uniform channel of arbitrary cross-section. J . Fluid Mech. 32, 353-365. PEREGRINE, D. H. (1971). A ship’s waves and its wake. J. Fluid Mech. 49, 353-360. PEREGRINE, D. H. (1972). River currents and trains of waves. Bull. Znst. Math. Its Appl. 8, 326328.
D.H.Peregrine
116
PEREGRINE, D. H. (1974). Surface shear waves. J. Hydraul. Div., Proc. Amer. SOC.Civil Eng. 100, 1215-1227. [Discussion in 101, 1032-1034 (1979.1 PEREGRINE, D. H.,and SMITH,R. (1975),Stationary gravity waves on non-uniform free streams: jet-like flows. Math. Proc. Cambridge Phil. SOC.77, 415-438. PETERS, A. S. (1966). Rotational and irrotational solitary waves in a channel with arbitrary cross-section. Commun. Pure Appl. Math. 19, 445-47 1. PHILLIPS, 0. M. (1959). The scattering of gravity waves by turbulence. J. Fluid Mech. 5, 177- 192.
PHILLIPS, 0.M. (1966). “The Dynamics of the Upper Ocean.” Cambridge Univ. Press, London and New York. PHILLIPS, 0. M. (1967). Theoretical and experimental studies ofgravity wave interactions. Proc. Roy. SOC, Ser. A 299, 104- 119. PHILLIPS, 0. M, and BANNER,M. L. (1974). Wave breaking in the presence ofwind drift and swell. J. Fluid Mech. 66, 625-640. RIBNER,H. S. (1957). Reflection, transmission and amplification of sound by a moving medium. J. Acoust. SOC.Amer. 29, 435-441. SARPKAYA, T. (1957). Oscillatory gravity waves in flowing water. Trans. Amer. SOC.Civil Eng. 125 564586.
SAVITSKY, D. (1970). Interaction between gravity waves and finite turbulent flow fields. Symp. Naval Hydrodyn., 8th, pp. 389-446. Off. Naval Res., Arlington, Virginia. (Discuss., pp. 44&447.) SCHWARTZ, L. W. (1974). Computer extension and analytic continuation of Stokes’ expansion for gravity waves. J. Fluid Mech. 6 5 553-578. SHEARER, J. R., and CROSS,J. J. (1965). Some aspects of the resistance of full form ships. Trans. Inst. Naval Architects 107, 459-473. (Discuss., pp. 486-501.) SHEMDIN,0. H. (1972). Wind-generated current and phase speed of wind waves. J. Phys. Oceanogr. 2,411-419. SHEPARD, F. P., EMERY, K. 0.. and LA FORD, E. C. (1941). Rip currents: a process of geological importance. J . Geol. 49. 337-369. SILCOCK,G. (1975). On the Stability of the Parallel Stratified Shear Flows. Ph.D. Thesis, Bristol, England. SLEATH, J. F. A. (1973). Mass transport in water waves ofvery small amplitude. J . Hydraul. Res. 11, 369-383. SLEATH,J. F. A. (1974). Mass transport over a rough bed. J. Mar. Res. 32, 13-24. SNODGRASS, F. E., GROVES, G. W., HASSELMANN, K. F., MILLER,G. R., MUNK,W. H., and POWERS,W. H. (1966). Propagation ofocean swell across the Pacific. Phil. Trans. Roy. SOC. London, Ser. A 259,431-497. STOKES, G. G. (1847). On the theory of oscillatory waves. Pans. Cambridge Phil. SOC. 8, 44-455. [Math. Phys. Pap. 1, 197-229 (1880).] TANIDA, Y., OKAJIMA, A,, and WATANABE, Y. (1973). Stability of a circular cylinder oscillating in uniform flow or in a wake. J. Fluid Mech. 61, 769-784. TATINCLAUX, J.-C. (1970). Effect of a rotational wake on the wavemaking resistance of an ogive. J. Ship Res. 14, 8499. TAYLOR,SIRGEOFFREY (1955). The action of a surface current used as a breakwater. Proc. Roy. SOC, Ser. A 231, 6 4 6 7 . TAYLOR, G. I. (1962). Standing waves on a contracting or expanding current. J. Fluid Mech. 13, 182-192.
THOMPSON, P. D. (1949). The propagation of small surface disturbances through rotational flow. Ann. N . Y . Acad. Sci. 51, 463-474. TSAO,S. (1959). Behavior of surface waves on a linearly varying current. T r . Mosk. Fiz.-Tekh. Inst. Issled. Mekh. Prikl. Mat. 3, 66-84. (In Russ.)
Interaction of Water Waves and Currents
117
TUNG,C. C., and HUANG,N. E. (1973). Combined elTects of current and waves on fluid force, Ocean Eng. 2, 183-193. H. G. M., and V A N WIINGAARDEN, L. (1969). Gravity waves over a non-uniform VELTHUIZEN, flow. J. Fluid Mech. 39, 817-829. VINCENT,C. E. (1975). The interaction between surface waves and tidal currents. Ph.D. Thesis, Southampton Univ., Southampton, England. J. V. (1973). The wave resistance of ships. Aduan. Appl. Mech. 13, 93-245. WEHAUSEN, WHITHAM,G. B. (1962). Mass, momentum, and energy flux in water waves. J. Fluid Mech. 12, 135-147. WHITHAM, G. B. (1965a). Non-linear dispersive waves. Proc. Roy. SOC.,Ser. A 283, 238-261. WHITHAM, G. B. (1965b). A general approach to linear and non-linear dispersive waves using a Lagrangian. J. Fluid Mech. 22, 273-283. G. B. (1967a). Variational methods and applications to water waves. Proc. Roy. Soc., WHITHAM, Ser. A 299, 6-25. WHITHAM, G. B. (1967b). Non-linear dispersion of water waves. J. Fluid Mech. 27, 399-412. WHITHAM, G. B. (1974). “Linear and Non-Linear Waves.” Wiley (Interscience), New York. WILLIAMS, J. A., and WIEGEL,R. L. (1962). Attenuation of wind waves by a hydraulic breakwater. Proc. Conf: Coastal Eng., 8th pp. 500-520. Wu, J. (1975). Wind-induced drift currents. J. Fluid Mech. 68, 49-70. YIH, C . 3 . (1969). “Fluid Mechanics.” McGraw-Hill, New York. YIH, C.4. (1972). Surface waves in flowing water. J. Fluid Mech. 51, 209-220.
NOTES ADDED IN PROOF BIRKEMEIER, W. A., and DALRYMPLE, R. A. (1975). Nearshore water circulation induced by wind and waves. Proc. Symp. Modeling Techniques, Amer. SOC.Civil Eng. pp. 1062-1081. Extends the work of Noda er al. (1974).
PURI,K. K. (1974). Waves on a shear flow. Bull. Aust. Math. SOC.11, 263-277. Unsteady linear waves on a flow U,,+ bz are calculated; the solutions may be of help in a more detailed study of critical layers (e.g., see p. 84). RUPERT,V. C. (1976). Wave current interaction in a.random wave field: a practical approach. J . Geophys. Res. 81, 363-367. This paper differs appreciably from those discussed on p. 53. SMITH,R. (1975). The reflection of short gravity waves on a non-uniform current. Proc. Cambridge Phil. Sac. 78, 517-525. An account of caustics which is complementary to that given in Sections 1I.D and 1I.E.
THOMSON, J. A,, and WEST,B. J. (1975). Interaction of small-amplitude surface gravity waves with surface currents. J. Phys. Oceanog. 5, 736-749. An interesting treatment of the effect of currents of the type described in Section I1,D.
This Page Intentionally Left Blank
Generation of Magnetic Fields by Fluid Motion H. K . MOFFATT Department of Applied Mathematics and Theoretical Physics University of Cambridge Cambridge. England
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 I1 . Magnetokinematic Preliminaries . . . . . . . . . . . . . . . . . . 125 A . Idealization of the Kinematic Dynamo Problem . . . . . . . . . . 125 B. Magnetic Field Representations . . . . . . . . . . . . . . . . . 127 C. Alfven's Theorem and Woltjer's Invariant . . . . . . . . . . . . . 128 D. Natural Decay Modes and Force-Free Fields . . . . . . . . . . . 129 111. Convection, Distortion. and Diffusion of B Lines . . . . . . . . . . . 130 A . Balance of Stretching and Diffusion in a Magnetic Flux Rope . . . . 131 B. Flux Expulsion by Flows with Closed Streamlines . . . . . . . . . 131 C. Topological Pumping of Magnetic Flux . . . . . . . . . . . . . . 133 D. Generation of Toroidal Field by Differential Rotation . . . . . . . 134 IV. Some Basic Results . . . . . . . . . . . . . . . . . . . . . . . . 135 A . Cowling's Theorem and Related Results . . . . . . . . . . . . . 135 B. Rotor Dynamos . . . . . . . . . . . . . . . . . . . . . . . . 137 V . The Mean Electromotive Force Generated by a Random Velocity Field . 139 A . The Two-Scale Approach . . . . . . . . . . . . . . . . . . . . 139 B. The Strong Diffusion Limit . . . . . . . . . . . . . . . . . . . 141 C. Evaluation of all and But for a Random Wave Field . . . . . . . . 145 D. Effect of Turbulence in the Weak Diffusion Limit, 1-0 . . . . . . 147 E. The Forms of ail and /Iuh in Axisymmetric Turbulence . . . . . . . 150 F. Dynamo Equations for Axisymmetric Mean Fields Including Mean 152 Flow Effects . . . . . . . . . . . . . . . . . . . . . . . . . VI . Braginskii's Theory of Nearly Axisymmetric Fields . . . . . . . . . . 154 A . Lagrangian Transformation of the Induction Equation . . . . . . . 154 B. Nearly Axisymmetric Systems . . . . . . . . . . . . . . . . . . 155 C. Nearly Rectilinear Flows; Effective Fields . . . . . . . . . . . . . 157 D. Dynamo Equations for Nearly Rectilinear Flows . . . . . . . . . 158 E. Comments on the General Approach of Soward . . . . . . . . . . 160 F. Comparison between the Two-Scale and Nearly Axisymmetric 161 Approaches . . . . . . . . . . . . . . . . . . . . . . . . . VII . Analytical and Numerical Solutions of the Dynamo Equations . . . . . 163 A . The a2 Dynamo with a Constant . . . . . . . . . . . . . . . . 163
119
120
H. K. Moflatt
B. a’ Dynamos with Antisymmetric a . . . . . . . . . . . . . . . . C. Local Behavior of am Dynamos . . . . . . . . . . . . . . . . . D. Global Behavior of am Dynamos . . . . . . . . . . . . . . . . VIII. Dynamic Effects and Self-Equilibration . . . . . . . . . . . . . . . A. Waves Influenced by Coriolis Forces, and Associated Dynamo Action B. Magnetostrophic Flow and the Taylor Constraint . . . . . . . . . C. Excitation of Magnetostrophic (MAC) Waves by Unstable Stratification D. Mean Flow Equilibration . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
164 165 166 168 168 170 171 174 176
I. Introduction The existence of the magnetic field of the Earth, and its variation with time, presents a profound challenge to geophysics. This field, though influenced slightly by electric currents in the ionosphere, is predominantly of internal origin and is associated with a large-scale azimuthal current distribution in the liquid core of the Earth. It is well known (see, e.g., Hide and Roberts, 1961) that the temperature of the core is far above the critical value (the “Curie point”) at which permanent magnetization can persist. Moreover, in the absence of any regenerative action, the electric currents in the core would decay through ordinary resistive (“ohmic”) dissipation in a time of order 104-105 years. Geomagnetic studies indicate, however, that the Earth’s field has existed in one form or another for at least lo8 years and is probably as old as the Earth itself, and further that, although the main dipole field exhibits random rapid reversals, a phenomenon reviewed by Bullard (1968), it remains at least quasi-steady between reversals for periods up to order lo6 years, i.e, one or two orders of magnitude greater than the natural decay time. It is now generally agreed that this persistence of the Earth’s field can only be explained in terms of electromagnetic induction, whereby the electric currents that provide the field are generated by motion of the fluid in the core across the self-same field, which permeates the core region as we11 as the nonconducting exterior. The characteristic feature of such dynamo action is that the field is maintained exclusively by the action of the fluid velocity and without the help of any external source of field. This type of behavior is most simply illustrated in terms of the simple disk dynamo illustrated in Fig. 1. The electrically conducting disk rotates about its axis with angular velocity Q and a conducting wire makes sliding contact with the rim of the disk and with its axis, as shown; the wire is twisted into a circle in its passage from the rim to the
Generation of Magnetic Fields by Fluid Motion
12 1
U FIG.1. The self-exciting disk dynamo; note particularly the concentrated shear at the sliding contact and the lack of reflectional symmetry of the system.
axis in such a way that any current flowing in the wire gives rise to a magnetic field with nonzero flux across the disk. The rotation of the disk in the presence of this flux generates a radial electromotive force, and since a closed circuit is available, a net current I ( t ) flows along the wire. The flux then equals MI, where M is the mutual inductance between the wire and the rim of the disk, and I is given simply by
L dlldt + RI
= MRI,
(1.1)
where L and R are, respectively, the self-inductance and resistance of the complete current circuit. Clearly, if R < MQ the current grows, and if R is maintained at a constant value, this growth is exponential. The state with I = 0 is then unstable to the growth of small electromagnetic disturbances. Such growth cannot, of course, continue indefinitely. The Lorentz force j A B (where j is the current distribution in the disk and B the magnetic field) provides a net resisting torque MI’, and (1.1) must be coupled with the equation of angular motion of the disk C dRldt = G - MI’,
(1.2)
where C is the moment of inertia of the disk about its axis, and G the applied torque. If G (rather than R) is kept constant, then as I increases, R decreases until an equilibrium is reached in which G = MI’,
R = RIM.
(1.3)
Note that the angular velocity of the disk in this equilibrium state does not depend on the appiied torque ! This simple dynamo relies for its success on the carefully contrived path that the current is forced to follow. The conductor (disk + wire) occupies a region of space that is not simply connected, a feature that is of course not shared by the conducting core of the Earth. There are, however, two features of the disk dynamo configuration that deserve particular emphasis, because
H . K . Moflatt
122
these features do recur in the fluid context and are both intimately associated with successful dynamo action. First, there is a region of concentrated shear at the sliding contact on the rim of the disk; the counterpart of this in the fluid context is differential rotation, which plays an important part in generating toroidal magnetic field from poloidal magnetic field (see Section 111,D). Second, the configuration lacks reflectional symmetry in that the sense of twist of the wire in Fig. 1 bears a very definite relation to the sense of the angular velocity of the disk: if the twist is reversed, then Mi21 in (1.1) is replaced by - MRZ, and rather than dynamo action we have accelerated decay of any transient current in the wire. This lack of reflectional symmetry also has its counterpart in homogeneous fluid systems. The simplest measure of the lack of reflectional symmetry of a localized fluid motion u(x) is its helicity
a quantity that admits interpretation in terms of the degree of linkage (or " knottedness") of its constituent vortex lines (Moffatt, 1969). We shall describe in the following sections (particularly Section V) how the presence of helicity is of vital importance for the process of regeneration of poloidal field from toroidal field, i.e., for the closing of the dynamo cycle that makes field regeneration a reality. The Earth is, of course, not the only celestial body that exhibits a significant large-scale magnetic field. Among the planets, Jupiter, Mars, and Mercury are now known to have this property also; the radii, rotation rates, and dipole moments of these planets in comparison with the Earth are displayed in Table 1 (from Dolginov, 1975). A theory that successfully explains the Earth's field may clearly have relevance in the context of these TABLE 1 COMPARATIVE FIGURES FOR THE EARTH, JUPITER,' MARS:
AND
MERCURY~
~~
Radius R (km) Earth Jupiter Mars Mercury a
6371 7 1,351
3386 2439
Dipole moment p (G km3)
P/R3 (GI
Rotation period (days)
8.05 x 10" 1.31 x 1015 2.47 x 107 4.8 x 107
3.11 x lo-' 3.61 6.36 x 3.31 x 10-3
1 0.415
Warwick (1963);Smith et a/. (1974).
* Dolginov et a/. (1973). Ness et al. (1974).
1
58
Generation of Magnetic Fields by Fluid Motion
123
other planetary fields. The internal constitution of Mercury, Mars, and Jupiter is, of course, largely a matter of speculation at present; it may be that detailed observation of the surface magnetic fields of these planets will in the long run provide (via theoretical argument) information about their interiors. In the case of Jupiter, it has been argued (see, e.g., Hide, 1974) that the core consists of a liquid alloy of metallic hydrogen and helium under high pressure and that this core provides the seat of magnetohydrodynamic dynamo action. The magnetic field of the Sun (which is believed to be typical of “cool” stars with convective outer envelopes) is much more complex in its structure and behavior than that of the Earth, and it too is widely (though not universally) believed to be the result of dynamo action involving the two features, differential rotation and motions lacking reflectional symmetry, described above. Paradoxically, although the Sun is certainly remote as compared WKi-the Earth’s liquid core, we have much more detailed information about its magnetic field, simply because it may be detected and measured at its visible surface, i.e., at the surface of the convective region where, from a magnetohydrodynamic point of view, all the interesting action takes place. In the case of the Earth, we have no prospect or hope of any direct measurement of the magnetic field, or indeed of any other quantity, either in the liquid core or on its surface, and we must make do with what we know of the field on the surface of the solid Earth, a pale shadow of the field of the deep interior, and a dim and distant reflection of the fluid turmoil in that deep interior which is at the heart of our problem. The Sun’s field is characterized by active regions and by a weak general field near the north and south poles of its axis of rotation. Active regions are associated with strong upwelling from the convective envelope with an associated vertical stretching of any magnetic field lines that are convected upward. When this stretching is particularly localized and intense, the strong vertical field that is created (of the order of thousands of gauss) can locally suppress thermal convection; the resulting decrease in heat transport leads to a local cooling of the surface; radiation from this local region (of the order of hundreds of kilometers in horizontal extent) is largely suppressed, and in consequence it is seen from the Earth as a dark spot on the surface of the Sun. Such sunspots occur in pairs, roughly along a line of latitude, but with the leading spot (i.e., that to the East) slightly nearer the equatorial plane. Sunspot activity has been followed for more than 300 years and is known to follow a roughly periodic cycle with half-period of about 11 years. At the beginning of a sunspot cycle, pairs of spots appear within the band of latitudes about & 30” from the equatorial plane, with statistical symmetry about this plane, first at the higher latitudes only, then gradually over a wider band of latitudes that drifts, as the cycle proceeds, toward the equatorial plane. In
124
H. K. Mofatt
any pair of sunspots, the vertical magnetic field is positive in one and negative in the other; if positive in the westerly spot, the pair has positive polarity, otherwise negative. In any half-cycle of 11 years, all sunspot pairs in the northern hemisphere have the same polarity, and all in the southern hemisphere have the opposite polarity. In the following half-cycle, these polarities are reversed. The weak polar field of the Sun also follows a somewhat irregular periodic evolution with approximately the same period as that of the sunspot cycle. The field was first measured by direct magnetograph measurements in 1952 (Babcock and Babcock, 1955) and it has been followed closely since that date. The field around the north pole reversed in 1958 and again in 1971;the field around the south pole reversed in 1957 and again in 1972! At each reversal, for about one year, the fields at north and south poles therefore had quadrupole rather than dipole symmetry about the equatorial plane. The reversals apparently occur during that part of the sunspot cycle when sunspot activity is at its maximum. These observations are compatible with the following qualitative picture (essentially conceived by Parker, 1955a): the global magnetic field of the Sun includes poloidal and toroidal ingredients that are not steady but vary periodically in time, with period approximately 22 years. The poloidal field can be observed and has the polar reversal behavior described above; the toroidal field is contained in some way beneath the visible surface of the sun and cannot be directly detected. This toroidal field is coupled with the poloidal field and in a typical half-period drifts like a wave from polar regions toward equatorial regions, intensifying as it progresses. When this field reaches a certain critical level of intensity, local upwelling instabilities may develop in which ropes of toroidal flux are stretched vertically upward, breaking through the visible surface of the sun and forming sunspots as described above. As the buoyant fluid rises through several scale-heights, it expands due to the decreasing ambient pressure; as a result of the tendency to conserve angular momentum, the rising blob develops a rotation (the sense of rotation being such that it has negative helicity in the northern hemisphere, positive in the southern): hence the twist of the sunspot pair from the original line of latitude of the underlying toroidal field. As the periodic evolution proceeds, the toroidal fields of opposite signs from the two hemispheres interpenetrate and annihilate each other in the equatorial zone and the sunspot activity in consequence dies away. The process then repeats itself, the toroidal field again growing from polar to equatorial regions (but with a complete change of polarity from one half-cycle to the next). What part does the weak poloidal field play in this process? It just must be present, for otherwise the dynamo cycle cannot proceed. On the one hand, all the little eruptions over the solar surface generate a field with a
Generation of Magnetic Fields by Fluid Motion
125
radial component (i.e., poloidal field); since the surface eruptions are most intense in equatorial latitudes, one might expect the poloidal field to be most evident in these regions also. However, poloidal field can be redistributed by large-scale meridional circulation in the convective zone and this must presumably play a part in sweeping poloidal field back to the polar regions. Meridional circulation also tends to generate differential rotation (conservation of angular momentum again) and this differential rotation is the means by which the toroidal field is regenerated from the poloidal. These complicated interactions may seem far removed from the simplicity of the disk dynamo described at the outset; yet the two features-differential rotation and lack of reflexional symmetry-appear in the solar context as vital ingredients in its periodic behavior; the lack of reflexional symmetry appears in the rising, twisting blobs, described by Parker (1955a) as “cyclonic events,” and directly responsible for sunspot formation. The above description is, of course, purely qualitative and suggestive. In the sections that follow, we shall endeavor to show how the various physical ideas implicit in the description may be placed on a secure mathematical foundation, and to relate the various approaches to dynamo theory that have made progress in this direction over the last 20 years. The reader who wishes further background material in the terrestrial and solar contexts may consult a number of review articles that have appeared in recent years (Parker, 1970a; Roberts, 1971; Weiss, 1971, 1974; Roberts and Soward, 1972; Vainshtein and Zel’dovich, 1972; Moffatt, 1973; Gubbins, 1974) and the very extensive detailed references that these articles contain.
11. Magnetokinematic Preliminaries A. IDEALIZATION OF THE KINEMATIC DYNAMO PROBLEM Suppose that fluid of uniform electrical conductivity 0 is confined to a simply connected region of space V inside a closed surface S, and suppose that the region p exterior to S (extending to infinity) is nonconducting. Let u(x, t ) be the velocity field in V, satisfying
and let p(x, t ) be the density field satisfying the equation of mass conservation appt
+v
(pu) = 0.
(2.2)
126
H. K. Mofatt
For many purposes it will be sufficient to restrict attention to incompressible fluids of uniform density for which p=po, V.u=O. (2.31 Let j(x, t), B(x, t), and E(x, t) denote electric current, magnetic field, and electric field, respectively. Neglecting displacement current (certainly valid for phenomena on the long time scales considered), the equations relating j, B, and E in V are
V*B=O, poj = V A B =poa(E + UAB), aB/at = - V where p o = 411 x
(2.4) (2.5)
E,
(2.6) S.I. units. In p, where j = 0, B is determined by A
V*B=O, VAB=O. (2.7) Moreover, both normal and tangential components of B must be continuous across S, i.e., [n*B]: =0, [ ~ A B ] != O on S, (2.8) where n is the unit outward normal on S. Finally, we require that B be without singularities in V and in p, and that there should be no sources of magnetic field at infinity; this means that B must be at most dipole, i.e., 0 ( ~ - 3 ) as I = 1x1+ Elimination of j and E from (2.4)-(2.6) gives the well-known induction equation (which holds in V), aB/at = v A (u A B) + WB,
(2.9) where A = (p0a)-' is the magnetic diffusiuity of the fluid. For given u, we wish to explore the evolution of the field B as determined by (2.7)-(2.9) and the subsidiary conditions mentioned. A simple measure of the field level is the total magnetic energy M ( t ) = (2p0)-
j
B2 d 3 x.
(2.10)
V+P
If, for given u, M ( t )+ 0 as t -,co, then the motion u does not act as a dynamo. If M ( t ) f ,0 as t + 00, then the motion does act as a dynamo, there being ultimately sufficient rate of generation of magnetic energy by fluid motion to counteract the natural decay of magnetic energy due to ohmic dissipation associated with the finite conductivity of the fluid. In a complete theory that takes account of the dynamics of the fluid motion, u is, of course, constrained to satisfy the Navier-Stokes equation
127
Generation of Magnetic Fields by Fluid Motion
(with Coriolis forces, Lorentz forces, buoyancy forces, etc., included if the context so requires). In a purely kinematic approach at the outset, it proves useful to widen the scope of the investigation and to imagine that u is any kinematically possible velocity field (without dynamical restriction); the influence of dynamic constraints, which will be considered in Section VIII, is more easily comprehended after close investigation of the kinematic problem. B. MAGNETIC FIELDREPRESENTATIONS
1. Poloidal and Toroidal Decomposition Any solenoidal field B may be expressed as the sum of a poloidal ingredient B, and a toroidal ingredient B,, where
B, = v h v h ( X ~ ( X ) ) ,
B, = v A (xT(x)); S and T are the defining scalars for these fields. Note that
(2.11)
VAB, = VAVA(XT)
(2.12)
is a poloidal field with defining scalar T, while
V A B, = - V2(V A xS)= - V A (xVZS)
(2.13)
.
is a toroidal field with defining scalar -VzS. Note further that x & = 0, i.e., the lines of force of the % field lie on spheres r = const, as do the Hnes of force of V A %, a property that makes the representation particularly useful when problems with spherical boundaries are considered.
2. Axisymmetric Fields A field B is axisymmetric about an axis Oz when its defining scalars Sand T are independent of the azimuth angle 4 about 02. If S = S(r, 8), T = T(r, 8),where 8 is measured from 02, then
where
x
= - r sin 8 asps,
B6 = -a T@l.
(2.15)
x, theflux function, is the analog of the Stokes stream function for solenoidal velocity fields, and the lines of force of the B, field are given by x = const. The B, field may also be expressed in the form B, = V A = X/r sin 8, and i4 is a unit vector in the 9 direction.
A
(A&), where
H. K. Moffatt
128
3. Two-Dimensional Fields
It is frequently illuminating to consider configurations in which B depends only on two artesian coordinates, say x and y. In this case, the representation analogous to the above is B = B, B,, where now
+
and the BP lines are given by A = const. C. ALFVBN'STHEOREM AND WOLTJER'S INVARIANT It is an immediate consequent of (2.4)-(2.6) that, if @(t) is the flux of B across any surface spanning a closed curve C(t) that moves with the fluid, then
(E + u A B) dx = -
d@/dt = 'cw
a-'j
*
dx,
(2.17)
'cw
so that, in the perfect conductivity limit (a = co), @ is constant for any material curve C(t).It follows that in this limit B lines are frozen in the fluid, and in an incompressible flow, stretching of the B lines leads to proportionate intensification of the B field. Closely associated with the " frozen-field " concept is the invariance of the integral
4"
H - A .Bd3x,
(2.18)
(Woltjer, 1958). Here A is the vector potential of B, i.e., B = V A A, and it is supposed that B is a localized field so that the integral exists. The volume V in (2.18) may be any volume bounded by a material surface S on which B II = 0. The interpretation of H, is identical with that for the helicity integral (1.4) (which is likewise constant whenever circumstances are such that vortex lines move with the fluid), i.e., HM is a measure of the degree of topological complexity of the field B within the surface S-and this measure cannot, of course, change when the B lines are frozen in the fluid. We may note in passing that the solution of (2.9) in the limit a = co (i.e., A = 0) may be expressed in Lagrangian variables in the form (due to Cauchy) Bi(x, t ) = Bj(a, 0) 8 x i / a a j ,
(2.19)
where x(a, t ) is the position at time t of the fluid particle that was at position a at time t = 0.
Generation of Magnetic Fields by Fluid Motion
129
D. NATURAL DECAYMODESAND FORCE-FREE FIELDS If u is steady, i.e., u = u(x), then the problem (2.7)-(2.9) admits solutions proportional to exp( -pt), where possible values of p are determinate as eigenvalues of the problem. If, for all these values, R e p > 0, then B inevitably decays with time, while if for any eigenvalue Re p < 0, the corresponding field structure (the eigenfunction) grows exponentially in time until Lorentz forces modify the velocity field (cf. the rotating-disk situation discussed in the introduction). If p = p , + ipi then the condition pr = 0 is critical in that it determines the onset of dynamo action for the corresponding field structure Bp(x).If, when pr = 0, it also happens that p i = 0, then the resulting mode is steady under critical conditions; this is the sort of behavior that we look for in the context of the Earth's magnetic field, which is steady (with weak fluctuations) over very long periods. If, on the other hand, p i # 0 when pr = 0, then the resulting mode is oscillating under critical conditions; this type of behavior would be relevant in the solar context. Of course, when u = 0, all the p's are real and positive; in the important case when the volume V is spherical with radius R, the eigenvalues are given by pnq = l R - ' x i q , (2.20) where xnqis the qth zero of the Bessel function J , , 1 / 2 ( x )The . structure of the corresponding fields for r < R are given, in the notation of Bullard and Gellman (1954), by
SF + iSy = V A V A [xr- '/'J*+ 112(A-'pn4r)eid],
(2.21) (2.22)
The field S0,e matches to a dipole field in the exterior region r > R, the field Sy matches to an axisymmetric quadrupole field, and so on. From (2.20) the time scale of decay of these modes of simple structure is O(R'A- ') (as can, of course, be anticipated from dimensional analysis). The natural decay modes are closely related to field structures that are force-free, i.e., for which the Lorentz force j A B everywhere vanishes. Such fields arise naturally in the context of the kinematic dynamo problem, and it will be useful to set out some of their properties here. First, for such fields there exists a scalar field K(x) such that (2.23) p; 'j = V A B = K(x)B, and, since V * j = V * B = 0, it follows that B.VK=O and j-VK=O, i.e., B lines and j lines lie on a surface K = const.
(2.24)
130
H. K. Mofatt
The simplest example of a force-free field, with K constant everywhere, is (in Cartesians) B = Bo(sin Kz, cos Kz, 0);
(2.25)
the property V A B = KB is trivially verified. The B lines lie in planes z = const and rotate in a left-handed sense with increasing z. The vector potential of B is just A = K- ‘B, so that A B = K-’B2 = K - ’ B i .
(2.26)
The magnetic helicity density A * B is thus uniform; the field structure (2.25) has “ maximal helicity ” (Kraichnan, 1973). This and other similar examples have the property that the j field extends to infinity. There are in fact no force-free fields for which j is confined to a finite volume and B is everywhere continuous and O(r- 3, at infinity (see, e.g., Roberts, 1967, p. 109). It is, however, possible (Chandrasekhar, 1956) to construct force-free fields in a sphere I/ that match smoothly onto currentfree fields in the exterior region P that do not vanish at infinity: let ~ ( re), = Ar-”’J3/2(~r) cos e,
(2.27)
and B = VA(XS) + K - ’ V A V A ( X S )
for r
-= R.
(2.28)
Then it may be readily verified that, since S satisfies the Helmholtz equation (V’ + K’)S = 0, the field (2.28) does satisfy V A B = KB. However, since j n = 0 on r = R, and since B is parallel to j, we must also satisfy B n = 0 on r = R (a condition that the decay modes do not satisfy) and this requires that J3,2(KR) = 0. The exterior field in P is purely poloidal and is given by B = K-’VAVA(XS), where
-
S = -Bo(r - R3/r2)cos 8,
Bo = iAR-”2 dJ,,,(KR)/dR, (2.29)
the latter condition ensuring smoothness across r = R.
111. Convection, Distortion, and Diffusion of B Lines
In this section, we consider certain particular solutions of the induction equation (2.9) when u(x) is prescribed and steady. The behavior is strongly dependent on the order of magnitude of the magnetic Reynolds number R, = uo l/A [where uo and 1 are, respectively, velocity and length scales characteristic of u(x)]. Moreover, great care must be exercised when the two
Generation of Magnetic Fields by Fluid Motion
131
limiting processes R, --t co and t -+ 00 are considered-the solution may depend in a most sensitive manner on the order in which these limits are taken. A. BALANCE OF STRETCHING AND DIFFUSION IN A MAGNETIC FLUXROPE
Equation (2.9) represents the evolution of B under the joint action of the stretching of magnetic field lines and of their diffusion relative to the fluid. A well-known steady solution of the equation in which these effects exactly balance exists when u is the uniform extensional straining motion given by u = (-ax, -ay, 2az),
a > 0.
(3.1)
+ y’)),
(3.2)
The steady solution of (2.9) is then
B = (0, 0, Bo exp( -a/1)(x2
representing a flux rope of gaussian structure aligned with the z axis. The field (3.2), in fact, provides the asymptotic solution of (2.9) as t --+ co (with 1 fixed and nonzero). The flux in the rope is nBo1/a, so that for given flux, Bo becomes very large when 1 is very small, under this type of persistent stretching. This type of motion may be expected (locally) to generate the strong vertical magnetic fields observed in sunspots, as described in the introduction.
B. FLUXEXPULSION BY FLOWS WITH CLOSED STREAMLINES The phenomenon of flux expulsion was first explicitly considered by Parker (1963) and Weiss (1966). Suppose we have a steady two-dimensional incompressible flow with stream function $(x, y) and suppose that the fluid is permeated at time t = 0 by a magnetic field that is uniform and in the plane of the flow. For t > 0 the flow distorts the field and diffusion, of course, also influences its behavior. If R, 4 1, diffusion dominates and the field perturbations remain small [in fact, O(R,) relative to the initial field]. If R , $- 1, the picture is very much more complicated. During an initial phase whose duration is O(Ri/2)l/uo(1 and uo being the scales introduced above), diffusion is negligible and the field perturbations grow in intensity to O(Ri/’) times the initial field. There is then an intermediate stage, which has been studied in a particular case by Parker (1966), and whose duration is O ( R ~ ~ 2 ) l / uduring ,, which diffusion causes the breaking off of closed flux loops in regions of closed streamlines; these flux loops slowly decay and disappear and there is an associated net reduction in the total flux threading
H . K . Moffatt
132
the region of closed streamlines. Finally, the field settles down to a steady state, in which the flux across any region of closed streamlines is exponentially small. The difference between limtdmlimA-.o and limAdolimtdm is quite striking in this context: the former procedure gives a field whose energy density increases without limit (as t’); the latter procedure gives a steady field whose energy density is generally less than that of the uniform field that we started with! The following simple proof that the flux across any region of closed streamlines must vanish (under the second limiting procedure) appears to be new. Using the representation (2.16a) for the magnetic field, (2.5) may be expressed in the simple form DA/Dt E aA/at + u V A = IV’A. (3.3) It is supposed here that there is no applied electric field, so that E, = -aA/at. Under steady conditions then, V * (uA) = u * V A = IV’A,
(3.4) and in the limit 1+ 0, u * V A = 0, so that A is constant on streamlines, or equivalently A = A($). We now adapt the argument of Batchelor (1956) (as applied to the vorticity equation) to the present context: let C be any closed streamline, and integrate (3.4) over the area enclosed by C. Since n * u = 0 on C (where n is normal to C), the left-hand side integrates to zero. This focuses attention on the effects of diffusion when I is small but not quite zero. The right-hand side, on integration, gives
-
I n V A ds = I dA/d$
0= $C
$ a$/an ds = I K c dA/d$,
(3.5)
C
where Kc is the circulation around C. It follows that in the steady state (which may, of course, take a long time to attain) dA/d$ = 0, i.e., A = const, i.e., B = 0 throughout the region of closed streamlines. A net flux of field across, say, a periodic array of eddies with closed streamlines cannot, of course, be simply eliminated by this mechanism. What happens, as is clearly demonstrated in the numerical solutions of Weiss (1966), is that the flux is concentrated into sheets of thickness O(R; ‘ I 2 ) at the boundaries between adjacent eddies. A horizontal row of eddies will concentrate vertical flux in this way at the vertical cell boundaries, whereas any horizontal flux will be expelled to the regions above and below the eddies. The above proof can be simply adapted to cover the corresponding axisymmetric situation when steady meridional circulation acts on an axisymmetric poloidal field. Again, if the relevant magnetic Reynolds number is large, the field is ultimately excluded from any region of closed
Generation of Magnetic Fields by Fluid Motion
133
streamlines. This result has important implications for dynamo theory: meridional circulation that is weak can be conducive to efficient dynamo action, but meridional circulation that is too strong merely expels poloidal field from regions of closed streamlines (i.e., from the whole fluid region for an enclosed flow) and this effect is bound to be counterproductive, as indeed demonstrated in the numerical studies of P. H. Roberts (1972).
C. TOFQLOGICAL PUMPING OF MAGNETIC FLUX A rather fundamental variant of the flux expulsion mechanism has recently been discovered by Drobyshevski and Yuferev (1974). This study was motivated by the observation that in steady Benard convection between horizontal planes, fluid rises at the center of the convection cells and falls at the periphery; the regions of rising fluid are therefore separated from each other, whereas the regions of falling fluid are all connected. If the fluid is permeated by a horizontal magnetic field, then a field line near the upper p:ane will be distorted to lie entirely in a region of falling fluid and will therefore tend to be convected downward, a tendency that may be resisted to some extent by diffusion. A field line near the lower plate cannot be distorted to lie everywhere in the disconnected regions of rising fluid and cannot therefore be convected upward. One would therefore expect a net tendency for flux to be transported toward the lower plate; the Benard layer should act as a valve, allowing horizontal flux to pass downward but not upward. The particular velocity field chosen by Drobyshevski and Yuferev to demonstrate. this effect is (in dimensionless form) u=(-sinx(l+$cosy)cosz,
- (1 + f cos x) sin y cos z, (cos x + cos y + cos x cos y) sin z), (3.6) for which the cell boundaries are square; the more realistic choice of hexagonal cell boundaries would complicate the analysis without greatly adding to the insight provided. The magnetic field in the absence of fluid motion (or equivalently at zero magnetic Reynolds number) is taken to be ( B o , 0, 0), i.e., uniform in the x direction, and it is supposed that the boundaries z = 0, n are perfectly conducting so that the flux nBo is trapped in the gap between them. The steady solution of (2.9) may be obtained as a power series in R, when R, + 1; what is of interest is the average of this field over the horizontal plane, which turns out to have the expansion 7Ri cos 22 + 7 R: (28 cos z - 3 cos 32) + O ( R i ) B(z) = Bo 1 + __ 48n2 240x
(
(3.7)
134
H . K . Moflatt
The asymmetry about z = 71/2 appears at the O(R:) level, and this term indeed shows the expected increase in flux in the lower half of the gap. This phenomenon is of potential interest in the solar context. Convection in the Sun’s outer layers is turbulent, but there are also fairly stable and persistent large-scale structures, reminiscent of Benard cells, that survive even in the presence of this turbulence. These cells do show a preferred tendency for fluid to rise in discrete regions and to fall in connected regions, and any toroidal flux permeating this region will in consequence tend to be transported downward. The relevant magnetic Reynolds number is R,, = us1JIe where us and 1, are scales characteristic of the large-scale structures and I, is an effective eddy diffusivity associated with the smallscale turbulence (see Section V); R,, will probably be of order unity (even though the magnetic Reynolds number based on molecular diffusivity is very large). The phenomenon of flux pumping has been further investigated by Proctor (1975), who has demonstrated that when R, G 1, pumping can occur even when the topological distinction between upward- and downwardmoving fluid is absent. Proctor analyzes the effect of two-dimensional motions in detail and shows that a lack of geometrical symmetry about the midplane is sufficient to lead to a net transport of flux either up or down; e.g., if (w3)# 0, where w is the vertical velocity at the midplane and the angular brackets denote a horizontal average, then there will be a net transport, which Proctor describes as geometrical (as opposed to topological) pumping. When R, % 1, however, he demonstrates that this type of twodimensional geometrical pumping is almost nonexistent, whereas in this limit the Drobyshevski and Yuferev mechanism may be expected to be most effective (although no detailed analysis has yet been carried out).
D. GENERATION OF TOROIDAL FIELD BY DIFFERENTIAL ROTATION We conclude this section with a brief discussion of the process by which toroidal field is generated from poloidal field by differential rotation. Physically it is clear that differential rotation about an axis will tend to distort the field lines of an initially poloidal axisymmetric field B, if the angular velocity w varies along the field lines. In fact, if u = w(s, z)k A x, where (s, 4, z ) are cylindrical polar coordinates and k a unit vector along Oz, and if B = B,(s, z ) + BJs, z, t)i4, then the 4 component of (2.9) is For the moment we shall suppose that B, is maintained steadily by some unspecified mechanism. As expected, it is the gradient of w along B, lines that gives rise to generation of toroidal field. If I is small, and if initially
Generation of Magnetic Fields by Fluid Motion
135
B4 = 0, then B4 grows linearly with time until = O(R,)lB,I, at which stage a steady state is established. There is no question of flux expulsion in this case since B, is axisymmetric (if B, included any nonaxisymmetric ingredients then these would be expelled). The ultimate steady solution of (3.8) may be easily obtained if B,,and w are prescribed. For example, if B, = B, k where Bo is uniform, and if w = w(r)where r2 = s2 + z2, then the steady solution of (3.8) is r
B 4 = - -:B, (Lr3)-’ sin 8 cos 8 r;lw(rl) d r l .
jo
(3.9)
Note that B4 as given by this solution is antisymmetric about the “equatorial” plane 8 = 4 2 , and that if o ( r l ) decreases sufficiently rapidly as rl -+ co for the integral in (3.9) to converge, then B4 =.O ( r - 3 )as r + a. Contrast the situation when B, is an irrotational field with uniform gradient, i.e.,
B,
+ zk),
= C(-2si,
(3.10)
where is is a unit vector in the s direction. The steady solution for B6 then has two ingredients proportional to sin 0 and ~ P ~ ( c0)/88, o s but the former ingredient dominates for large I , and again provided o ( r l ) decreases sufficiently rapidly as rl -+ co, the asymptotic behavior of B4 for large r is
B4
-
(2C sin 8/3Lr2)
1
03
r:w(rl) dr,.
(3.11)
JO
This field is symmetric about 8 = 4 2 , and O(r-’) at infinity. In general, therefore, if a poloidal field B, is weakly varying in a region of differential rotation, then it is the local gradient of the field (rather than the local field itself) that determines the toroidal field generated at remote points when conditions are steady.
IV. Some Basic Results
A. COWLING’S THEOREM AND RELATEDRESULTS The impossibility of steady maintenance of an axisymmetric magnetic field by motions axisyrnmetric about the same axis (Cowling, 1934) is so well known as hardly to require comment here. The modifications and extensions of the theorem are numerous, but all reflect the basic fact that axisymmetric meridional circulation can redistribute poloidal flux but cannot systematically regenerate it. The equation for the flux function ~ ( r8), under
H. K. Moflatt
136
axisymmetric conditions [analogous to (3.35)] is
axlat + 4 - vx = A D ~ X ,
(4.1)
where 4 is the meridional velocity (assumed axisymmetric) and D2 the Stokes operator. The structure of this parabolic equation essentially ensures (Braginskii, 1964a)that V Xeverywhere tends to zero. Even if 1is nonuniform but satisfies merely the natural condition up V 1 = 0, standard manipulation of (4.1) and the relevant boundary conditions leads to this same conclusion (yet another minor extension of Cowling’s celebrated result !). Of course, when ( V x ( and so lBPl have reached a negligibly weak level, the toroidal field B+ must likewise decay, again essentially because of the parabolic structure of (3.8). An interesting variation on Cowling’s theorem has been claimed by Pichakhchi (1966). This is that steady dynamo action is impossible if the electric field E vanishes everywhere. (In the axisymmetric case with B, = 0, this is just a rewording of Cowling’s theorem since E I 0 in this situation under steady conditions.) If a steady dynamo with E = 0 were possible, then the field B would [from (2.5)] satisfy B * (V A B)= 0, i.e., it would have zero helicity everywhere and a correspondingly simple topological structure. The fact that this is apparently not possible is of course significant. There is also a counterpart of Cowling’s theorem for two-dimensional velocity and magnetic fields (Zel’dovich, 1957; Lortz, 1968). In this case, (3.3) is the governing equation, and by standard manipulations, provided A = o(r- I ) at infinity,
.
(dldt)
55
‘A2dx d y = -21
55 (VA)’
d x dy.
(4.2)
It follows that a steady state is possible only if VA = 0, i.e., only if B, = By= 0. Again, this removes the source of any possible regeneration of B,, which must also therefore vanish in a steady state. If u and B are stationary random functions (with zero mean) of x and y, then (4.2) must be replaced by d ( A 2 ) / d t = -2A((VA)’),
(4.3)
where ( ) indicates averaging over the x-y plane. No matter how small 1 may be, this again implies ultimate decay of the magnetic energy density. In the limit 1 + 0, ( A z ) apparently remains constant; however, this holds only so long as ((VA)’) remains finite; in fact, ( ( V A ) * ) increases (just as in the particular situation described in Section III,B) until it is O(A-’); at this stage and the inexorable decay of the field the length scale of the field is 0(1’/2) then sets in.
Generation of Magnetic. Fields by Fluid Motion
137
The impossibility of dynamo action under purely toroidal motion (Bullard and Gellman, 1954; Backus, 1958) is also closely related to Cowling’s theorem, in that it follows from the equation
D(x B)/Dt = (B * V)(X
*
U)
+ rZVZ(x
B),
(4.4)
.
which may be derived from (2.9) together with V u = 0. If u is purely toroidal, then x u = 0 and so x B inevitably decays everywhere to zero. The decay of Bp and is almost an immediate consequence. Busse (1975) has on the basis of Eq. (4.4) obtained a necessary condition (in terms of a lower bound on the poloidal velocity) that must be satisfied for successful dynamo action. B. ROTOR DYNAMOS In view of the various antidynamo theorems described above, it was of course of crucial importance that the possibility of steady dynamo action in a fluid of uniform conductivity occupying a simply connected domain be unambiguously established for at least one kinematically possible velocity field (no matter how artificial from a dynamical point of view). Until this was done (Herzenberg, 1958) it was by no means clear that a master theorem proving the absolute impossibility of such dynamo action might not at some stage be proved. Herzenberg’s dynamo consisted of two spherical rotors (i.e., quasi-eddies) embedded in a fluid sphere, the conductivity being uniform throughout. In fact, it is easier to comprehend the three-rotor problem (Fig. 2) considered? by Gibson (1968). Suppose that three spheres S,, S, , S, each of radius a, with centers located at the points (d, 0, 0), (0, d, 0), and (0, 0, d), where d 9 a, rotate with angular velocities (0, 0, -a),( -0, 0, 0), and (0,-o,0), respectively, where o > 0; the conductivity c throughout the whole space is assumed uniform. Note immediately the lack of reflectional symmetry in this configuration. The principle of the dynamo is roughly as follows: suppose that there are nearly uniform fields of the form B, x (0, 0, B), B, x (B, 0, 0), and B, x (0, B, 0) in the neighborhoods of S , , S 2 , and S , ,respectively. Then we have the possibility of a cyclic generation in which the toroidal field generated by rotation of S, (a = 1, 2) acts as the “applied” field Be+ in the neighborhood of S,+ and the toroidal field generated by rotation of S , acts as the applied field B, in the neighborhood of S,. The subtlety of the problem derives from the fact that, as mentioned at the end of Section III,D, the fields generated by differential rotation in each case are determined to an important degree by the local field gradient as well as by the local field itself, and this has to be taken into account in the
,
,,
t The particular configuration considered here was also discussed by Venezian (1967).
138
H. K . MofSatt
FIG.2. The three-rotor dynamo of Gibson (1968). The spheres rotate as indicated and generate toroidal fields B,, which act as the applied poloidal fields for S , (a = 1, 2, 3).
detailed calculation. The condition obtained by Gibson for steady dynamo action (adapted to this particular configuration) is R , = wa2/A= l O f i ( d / ~ ) ~ ,
(4.5)
correct to leading order in the small parameter a/d. A working dynamo based on the interaction of two rotors has been constructed in the laboratory by Lowes and Wilkinson (1963, 1968).The rotors are cylinders inclined to each other and embedded in a block of material of the same conductivity, electrical communication between the rotors and the block being provided by a lubricating film of mercury. Not only was dynamo action demonstrated with this model [through observation of a sudden large increase in the local magnetic field when the angular velocities of the cylinders were increased beyond a certain critical value-analogous to that given by (4.5)], but also reversals of the field were observed when the dynamo was functioning in the fully nonlinear regime-an observation of the greatest interest in view of the known random reversals of the Earth’s magnetic field mentioned in the introduction. A further ingenious example of dynamo action associated with a pair of rotors has been analyzed by Gailitis (1970). In this case, the rotors are toroidal rather than spherical, and the velocity field is axisymmetric about the common axis of the two toruses. The magnetic field that is maintained is, however, nonaxisymmetric, and there is no conflict with Cowling’s theorem.
Generation of Magnetic Fields by Fluid Motion
139
V. The Mean Electromotive Force Generated by a Random Velocity Field
A. THETWO-SCALE APPROACH Motion in the convective zone of the Sun is certainly turbulent, and any dynamo theory that fails to take account of this fact is hardly realistic. Likewise, motion in the core of the Earth almost certainly consists of a mean and a random ingredient. It is not clear whether the random ingredient is turbulence in the normal sense of the word, or rather a random field of waves influenced by Lorentz, Coriolis, and buoyancy forces; from a purely kinematic point of view, this distinction is not crucial, and we merely assume throughout this section that u(x, t) is a stationary random function of both x and t with zero mean-effects of nonzero mean velocity will be considered in Section VI. We are primarily concerned with the evolution of the mean magnetic field, which may be supposed to have a characteristic length scale L large compared with the scale l that characterizes the “background” velocity fiel’d u; in the case of turbulence, 1 will be the scale of the energy-containingeddies (Batchelor, 1953), while in the case of random waves, 1 will be, say, the wavelength of the most energetic modes represented in the spectrum of u. Either way, we can average the induction equation (2.9) over a spatial scale intermediate between 1 and L to obtain
dB,/dt = V A B
+ IV’B,
,
(54
where B, = (B), B = B, + b, and d = (u A b). This two-scale approach was introduced by Steenbeck et al. (1966) and has since had a revolutionary impact on the subject?. A series of papers by these authors, developing their approach to “mean field electrodynamics” has been collected together in English translation by Roberts and Stix (1971). The main problem, of course, analogous to the closure problem of turbulence dynamics, is to find a means of expressing d in terms of B, so that (5.1) may be integrated. The task is, however, easier because the basic equation for B is linear, albeit with a random coefficient. The equation for b, obtained by subtracting (5.1) from (2.9), is
db/dt = VA(UAB,)+ VA(uhb - (uhb))
+ IV’b,
(5.2)
and if we suppose that b = 0 at same initial instant t = 0, this establishes a linear relationship between b and B, , and so between 6” and B,; since the t The concepts of a mean electromotive force and of an associated eddy conductivity were already present in earlier work (e.g, Kovasznay, 1960).
H . K. Moflatt
140
scale L of Bo is very large, such a relationship may presumably be developed as a series di = q j & j
+ p i j k dBOj/dxk +
d2BOj/dxk 8x1
+
(5.3) the coefficients a i j , pis, . .., being pseudotensors determined in principle by the statistical properties of the u field and the parameter I (which, of course, plays an important part in the solution of (5.2) (pseudo because 8 is a polar vector, whereas Bo is an axial vector). It is important to note that these pseudotensors do not depend on B,; hence aij may be evaluated on the simplifying assumption that B, is uniform; then B i l k may be evaluated on the assumption that dBo,/dx, is uniform, and so on. Attention in most investigations has been focused on the first two terms of (5.3) on the grounds that it is essentially a series in ascending powers of l/L, and subsequent terms are likely to have negligible effect when L is large. There are, however, considerable subtleties here, particularly when I is very small, and the possible influence of subsequent terms perhaps deserves investigation. For the present, however, we truncate the series (5.3) after the second term and investigate some of the consequences. First, suppose that the u field exhibits no preferred direction in its statistical properties; since ail, flijk, are essentially statistical properties of the turbulence, they must then be invariant under rotations of the frame of reference and must therefore take the form a.. = ma.ij IJ
9
YijkI
pijk
=b i j k
9
“ ‘ 9
(5.4)
where a is a pseudoscalar and fl a pure scalar. Here the crucial role played by “lack of reflectional symmetry” comes into evidence. If the u field is reflectionally (as well as rotationally) symmetric in its statistical properties, then a (being a pseudoscalar that is not invariant under change from a rightto a left-handed frame of reference) must vanish. No such conclusion applies to the fl term. If the turbulence lacks reflectional symmetry, then the a term will in general be nonzero, and the relation between B and B, becomes
d = aB, - P A B , .
(5.5)
In view of the assumed statistical homogeneity of the u field, a and p are constants, and (5.1) becomes
aB,/at = aV A Bo ’+ ( I
+ p)V2Bo.
(54 Hence fl plays the role of a turbulent diffusivity; it is to be expected that 1> 0, although no general proof of this appears yet to be available. The a term, on the other hand, has quite a novel structure (from the point of view of conventional electrodynamics) and is in fact of crucial importance for dynamo theory.
Generation of Magnetic Fields by Fluid Motion The average of Ohm's law (2.5), incorporating (5.5), becomes J, = a(E, aB, - BV A B,),
+
141
(5-7)
where Jo = (j), E, = (E), or equivalently, ae = cr( 1 /?upo)-'. Jo = a,(E, aB,), (5.8) The a term therefore tends to drive mean current along the lines of mean magnetic field. In a spherical geometry, this effect in the presence of a toroidal field will generate a toroidal current, which acts as the source of a poloidal field. This therefore is the key to the means by which poloidal field may be regenerated from toroidal field by nonaxisymmetric random motions. The explosive character of Eq. (5.6)may be recognized very quickly if we suppose for the moment that our fluid fills all space and if we consider a magnetic field having an initial structure satisfying V A B, = KB, (i.e., one of the force-free fields of Section I1,D). For such a field, V'B, = -K2BQ, and so according to (5.6) the field will retain its spatial structure and develop exponen tially like exp cot, where o = aK - ( A + B)K2. (5.9)
+
+
Clearly we have exponential growth (i.e., dynamo action) if aK > ( A + B)K2
(5.10)
(and clearly we must choose K to have the same sign as a in order to ensure this possibility). Provided 1 K I is sufficiently small (i.e., provided the scale of Bo is sufficiently large), condition (5.10) is satisfied. Dynamo growth is then assured for force-free modes of sufficiently large length scale. It is, of course, important to obtain an explicit representation of aij and Bijk in terms of the statistical properties of the u field; this stage of the problem is analogous to the statistical mechanics problem of obtaining expressions for the various transport coefficients in terms of the statistical substructure of the medium considered. In the present context the substructure is provided by the background random velocity field. Unfortunately, determination of aij and Bijk is possible only in certain limiting situations; however, these limiting situations are in themselves of particular interest and will be considered in the following sections.
B. THESTRONG DIFFUSION LIMIT If R , = uol/A << 1, where u, = (u~)~'~,then the diffusion term AV2b in (5.2) clearly dominates over the " interaction " term
G = V A ( U A b - (UAb)),
(5.11)
H . K. Moffatt
142
which may be neglected to lowest order. The resulting equation
ab/at
=
v A (u A B,) + LV2b
(5.12)
is soluble by straightforward means, as recognized by Liepmann (1952) in a discussion of the spectrum of field fluctuations generated by turbulence in the presence of a uniform magnetic field. It is important now to distinguish between conventional turbulence whose characteristic time scale (the turnover time of the energycontaining eddies) is of order l/u,, and a random wave field whose time scale (the inverse of the dominant frequency) is determined by the relevant dispersion relation and is quite independent of I/#,. In the former case, 1 ab/at I is of the same order of magnitude as 1 G I and should, for consistency, be dropped also. Let us fist determine aij in this situation. As remarked earlier, to do this we may suppose Bo uniform, and (5.12) becomes simply
1V2b = -(Bo
*
V)u.
(5.13)
Let p(k, t ) and q(k, t ) be the space Fourier transforms of u and b, respectively. [The fact that p and q must be generalized functions (Lighthill, 1959) does not invalidate any of the operations that follow.] Then from (5.13),
- l k 2 q = - i(Bo k)p.
(5.14)
The spectrum tensor cPij(k) is related to p(k, t ) by (5.15) where an asterisk here and subsequently denotes a complex conjugate, and similarly for the spectrum tensor Tij(k) of the b field. The relation between these two spectrum tensors may be derived immediately from (5.14) (Golitsyn, 1960) in the form (5.16) More interestingly, the vector 8 may be obtained from (5.14) in the form 8, = aij B,,, where aij = isikll-
5 k- ’k,cPkl(k)d3k.
(5.17)
The Hermitian symmetry of Ok,(k) (Batchelor, 1953) guarantees that this expression for ail is real; moreover, the incompressibility condition in the form kifij(k) = 0 may be used to show that aij [as given by (5.17)] is also symmetric (Moffatt, 1970a).
Generation of Magnetic Fields by Fluid Motion
143
For turbulence that is statistically invariant under rotations (i.e., showing no preferred direction), @,,(k) takes the form (5.18) where E(k) ( 20 for all k ) is the energy spectrum function, and F(k) the helicity spectrum function, satisfying +(u2) =
5
m
W
E(k) dk,
(u * V A U )
=
jo F(k) dk.
(5.19)
0
The Schwarz inequality
(P * kAP*)2 5 < (PI2)< IkAP12)
(5.20)
may be translated into spectral terms to show that, for all k,
I F(k)I I 2kE(k).
(5.21)
It is clear that Only the antisymmetric part of @kl contributes to (5.17), and substitution of (5.18) in fact gives aij = adii, where a = -(1/3A)
J’
m
k-2F(k) dk.
(5.22)
0
Thus a is expressed as a weighted integral of the helicity spectrum function. IfF(k) is either positive for all k or negative for all k, then a and (u V A u) have opposite signs; in this case an order of magnitude estimate of a is given from (5.19b) and (5.22) in the form am
-@-‘12(u
*
a),
(5.23)
where o = V A U. The pseudotensor B i j k can be obtained by the same method and is likewise proportional to 1- It follows on dimensional grounds that all components of Bijk have order of magnitude at most I2-b;1’ = R i A; since R, -g 1, effects associated with B i j k are in this limit swamped by the molecular diffusion term 1V2Bo in (5.1) and may reasonably be ignored. The conclusion that ail is symmetric does not persist if the calculation leading to (5.17) is extended to higher powers in R,. The result (5.17) may be regarded as the leading term of an expansion of the form (5.24) n= 1
where the a!;) are dimensionless pseudotensors that may in principle be determined by iteration based on (5.2);a$)can by this means be expressed as
H.K. Moffatt
144
an n-fold weighted integral of an (n + 1)th-order spectrum tensor. Krause (1968) has considered the question of convergence of this type of expansion and concludes that it does converge for all finite R,, although the indications are that the convergence may be extremely slow for large R , . There are further indications from parallel work by G. 0. Roberts (1970, 1972) on spatially periodic dynamos that a!;) is symmetric or antisymmetric according as n is odd or even; this conjecture requires further investigation in the turbulence context. In general, however, it is clear that au may be decomposed into its symmetric and antisymmetric parts: (5.25)
Theeffect ofthe antisymmetric part in (5.1) is to provide a term V A (V A B,) on the right-hand side, where V is a velocity (uniform in so far as the turbulence is homogeneous) that merely convects the large-scale field pattern. In conjunction with the a effect, the force-free modes considered in Section V,A would then propagate relative to the fluid with phase velocity V. If there is also a mean fluid velocity U,then the effective mean velocity as far as the magnetic field is concerned is U + V; this concept of an “effective velocity” is a crucial ingredient of Braginskii’s (1964a) theory of nearly axisymmetric configurations (see Section VI). The appearance of an electromotive force with a component parallel to the ambient magnetic field [Eq. (5.5)] bears a simple physical interpretation, which was in fact given by Parker (1955a), who on the basis of inspired physical reasoning and heuristic argument anticipated the main lines of development of the subject by more than ten years! Consider the effect of a localized motion with positive helicity (a “cyclonic event ’* in Parker’s terminology). Such a motion tends to generate an 0-shaped loop in a line of force of an ambient magnetic field B, (Fig. 3a), and the loop is twisted so that its normal has a nonzero component in the original field direction.
I (a)
1 h)
FIG.3. Possible distortion of a line of force by a cyclonic event.
Generation of Magnetic Fields by Fluid Motion
145
When the twist is of limited amount (as it certainly will be in the strongly diffusive situation considered above) the loop may be conceived as being due to a current antiparallel to B,; a random superposition of such motions may then be expected to generate a mean current Jo antiparallel to B,, as implied by the result d = aB, with a < 0. This argument seems reliable both in the strong diffusion limit considered here and in the alternative situation considered by Parker when diffusion is weak but the events are so short-lived that the “limited twist” picture of Fig. 3a is applicable. In this weak diffusion limit, however, if the events are not short-lived, then a twist through 3x/2 (Fig. 3b) will give just the opposite effect from a twist through 4 2 , and this introduces some uncertainty into the sign of the effective value for a. Since the sign of a is of crucial importance for some of the dynamos described in Sections VI1,C and D it becomes important to have a reliable expression for a in the weak diffusion limit also (see Section V,D). C. EVALUATION OF aij AND
Bijk FOR A RANDOMWAVEFIELD
For a random wave field, it is natural to use a double Fourier transform in both space and time variables for u(x, t ) :
and similarly for b(x, t). The field has a random wave character (rather than a “ turbulent?’character) if the velocity amplitudes in each constituent wave are small compared with their phase velocities, i.e., provided I k3wii(k,w ) I + o / k . In this situation the interaction term G is again negligible in (5.2) and so b is determined by (5.12). Now, however, there is no reason to drop db/dt; moreover, (5.12) should be valid for all values of 1, both small and large, the case when R is small being now of greater potential interest. Note that in the limit of infinitesimal wave amplitudes, the constituent waves in (5.26) are noninteracting, with a dispersion relationship (that will be determined by dynamic considerations) w = o(k). For finite amplitude waves, however, nonlinear interactions will provide some forcing of waves at nonresonant frequencies, and w need not then be restricted by a dispersion relation; there may of course also be some extraneous forcing of waves. Taking the Fourier transform of (S.l2), with B,(x) a field whose second and higher space derivatives vanish, and constructing (u A b) as before leads
H.K. MofSatt
146
to the following expressions for aij and pijk: aij = i h i k r JJ (w‘
pijk= Re eiml
51
+ 1’k4)-’kakj@kl(k,o)dk d o ,
(w’
(5.27)
+ 1’k4)-’(iw+ Aka) (5.28)
where now alm(k,w ) is the double Fourier transform of the space-time velocity correlation tensor. Note that results for the strong diffusion limit (Section V,B) may be recovered by replacement of cPij(k, w ) by a r j ( k )d(o); in this limit, magnetic adjustment to velocity fluctuations becomes instantaneous. In the case when the wave amplitudes are isotropically distributed, (5.24) becomes aij = a d,, where m
00
a = +aii =
-*A
j-
do
j0
dk(w2
+ A2k4)- ‘k’F(k, w),
(5.29)
where F(k, 0) bears the same relation to cPkf(k,w ) as F ( k ) bears to cP,,(k)in (5.18). Similarly, (5.28) becomes pijk= fieijk, where
p = @ijk~ijk
= 41
J
OD
-m
doJ
m
dk(d
+ 1’k4)-’k’E(k,
0)
(5.30)
0
(Krause and Roberts, 1973; Roberts and Soward, 1975). Evidently j?is positive and again a has the opposite sign from F(k, w ) if this function is single-signed. It is noteworthy that both expressions (5.29) and (5.30) vanish in the perfect conductivity limit 1 40 [unless there is a divergence of the integrals in the neighborhood of w = 0, a possibility that can certainly be discounted if cPij(k,w ) = O(wz)as o -,01. The reason is essentially that in this limit b and u are in phase in each constituent wave, so that ii A 6* makes no contribution to (u A b). Dillon (1975) has shown that, as far as a is concerned, this result holds even when effects of the interaction terms G are included: by expanding in powers of the amplitude of the constituent waves, Dillon shows by induction that, to all orders, aii = 0 when 1 = O! The vanishing of a and p in the limit 1+ 0 does, however, depend critically on the assumption of a well-established wave field, with no transient
Generation of Magnetic Fields by Fluid Motion
147
effects present, and no zero frequency ingredients. If we adopt the alternative point of view and solve (5.12) with A = 0, B, uniform, and subject to the initial condition b(x, 0) = 0, we obtain
b(x, t ) = Bo * j:Vu(x, z) dz,
(5.31)
and so
and the relevant value of a when the field u is isotropic and t large is evidently
which need not vanish. So which expression is “correct”, (5.29) or (5.33)? If F(k, 0) # 0, then (5.33) is appropriate [and may in fact be obtained directly from (5.29) by replacing F(k, o)by F(k, 0) and integrating with respect to 013; if F(k, 0) = 0 however (i.e., if the wave spectral density vanishes as w + O ) , then the expression (5.29) is appropriate with the implication [consistent with (5.33)] thatcr-+OasA+O. This distinction is clearly important since a factor 1(or equivalently R; I ) in order of magnitude estimates of a makes a big difference when (as in the turbulent convection zone of the sun) R , % 1. The approach of Parker (1955a, 1970b) (see Section V,D) gives a result akin to (5.33) and corresponds to the limiting procedure limr-rmlimA+o;the approach of Braginskii (1964a,b) gives a result more akin to (5.29) and corresponds to the inverse procedure limA,o limt-+m.
D. EFFECTOF TURBULENCE IN THE WEAKDIFFUSION LIMIT,A + 0 For turbulence with typical time scale O(l/uo),there is no justification for the neglect of the interaction term G when A is small, and we must return to the exact Eq. (5.2). If we fix t > 0 and let 1+ 0, then the Cauchy solution (2.19) is valid and we may construct 8,= (u
A
b)i
= ( u A B ) ~= eijk(uj(a, t)B,(a,0 ) axk/aa,>,
(5.34)
where v(a, t) = u(x(a, t), t). If further we adopt the initial condition b(x, 0) = 0, then B,(a, 0) = Bo,(a, 0); and with the usual artifice that aij may
H. K. Moflatt
148
be calculated by assuming Bo uniform, we obtain immediately a i j ( t )= Eink(q,,(a,t ) dxkfdaj) = &imk
lot(u,(a, t ) auk(%
z)/aaj) dr. (5.35)
This tensor has a superficial resemblance with the diffusion tensor D i j ( t )for a scalar field (Taylor, 1922):
1
~ ~ ~= ( t
tir.
(5.36)
' 0
There are, however, several difficulties in the interpretation of ( 5 . 3 9 which may be catalogued as follows: (i) If the turbulence is statistically stationary in space and time, then the integrand in (5.36) is a function of t - z only and the integral certainly converges as t co. The integrand of (5.35) may, however, depend on t and t - r independently (since auk/da, is not a stationary random function of r-see Lumley, 1962) and there is no obvious guarantee that the integral converges as t + 00. (ii) If the integral (5.35) diverges or oscillates as t -,co (as is not implausible bearing in mind the discussion about loop twisting at the end of Section V,B), then the effects of weak molecular diffusion must in some way be reincorporated to force convergence in a time of order 12/A. This is extremely difficult, although possibly the approach of Saffman (1960) for the corresponding scalar problem might succeed. (iii) Even if the integral (5.35) does converge, there is no absolute guarantee that it gives a good approximation to the relevant value of a when I is small but nonzero. We have seen in Section V,C that expression (5.33) (which is certainly a convergent integral) may be quite misleading when R , is large but finite and t -+ co. The same may be true in the present context in which Lagrangian (rather than Eulerian) correlation tensors make a natural appearance. In the linearized context of Section V,C, transients certainly decay as I t -+ co ; it is not known whether the same is true when the interaction term G is retained. The question may be rephrased as follows: suppose u(x, t ) is a stationary random function of x and t, and b(x, 0 ) a stationary random function of x, both with zero mean, but with (u A b) # 0 at t = 0, and let b(x, t ) be determined by the exact induction equation with I # 0. Does (u A b) tend to zero or to infinity, or to something between as t co? [The related question of what happens to (b2) as t + 00 is an old one (Batchelor, 1950), which has been studied closely from many points of view (Schluter and Biermann, 1950; Moffatt, 1961, 1963; Saffman, 1963; Kraichnan and Nagarajan, 1967) but to which no clear-cut answer has yet emerged.] -+
Generation of Magnetic Fields by Fluid Motion
149
The expression (5.35) hears a close relationship with the expression obtained by Parker (1970b) on the basis of his cyclonic events model, namely,
(5.37) where X(a) is the displacement of the particle that starts at position a in an event, the angular brackets represent an average over events, and n is the number of events per unit volume per unit time. Expression (5.35)is preferable to (5.37)simply because (5.35) is defined (and therefore meaningful) for any turbulent velocity field, whereas (5.37)is defined only for velocity fields that can be represented in terms ofrandom cyclonic events; although such a representation is physically illuminating, it is unlikely that an arbitrary turbulent velocity field will in general admit such a representation. In the isotropic (“no preferred direction ”) situation, (5.35) becomes aij = a h i j , where
(5.38) and again the appearance of a helicity-type correlation (this time in terms of Lagrangian variables) is to be noted. It would be of great interest to be able to reexpress (5.38) in terms of standard Eulerian correlations (and undoubtedly an infinite series of these would be needed)-but the relationship between Lagrangian and Eulerian statistical quantities presents one of the notoriously difficult central unsolved problems of the theory of turbulence. The tensor Bijk(t)has an integral representation similar to (5.35)(Moffatt, 1974).In the isotropic situation this reduces to the form /Iijn = B(t)eijk,where ~ ( t=)
-i
1
2
0
0
j j{
-
- v(a, s,)V, - v(a, zl)))
+ i a ( t ) [‘a(z)dz,
*
v(a, 7,))
dzl dzz
(5.39)
0
where D(t) = &, and Dij and a(t) are given by (5.36)and (5.38), respectively. The last term of this expression ensures that if a(t)tends to a constant value as t -,00, then on this diffusionlesstheory p(t) is unbounded as t + 0O.t t Footnote added in proof: The functions a(t) and j ( t ) given by (5.38) and (5.39) have recently been computed by R. H. Kraichnan (private communication) for gaussian turbulence of maximal helicity; the results indicate that both a(t) and /I( tend ? to) constant nonzero values of order uo and uo l, respectively. This means that the divergence in the final term of (5.39) is compensated by an equal divergence in the double integral term. This interesting result may be peculiar to the assumed gaussian statistics.
H . K . Moflatt
150
Hence it is essential to retain the effects of molecular diffusivity in order to determine the appropriate form of @ in the weak diffusion limit when the turbulence lacks reflectional symmetry. This calculation (if it could be done) would inevitably lead to an expression of the form fi = uoIf(R,), where f(R,) --t co as R, co.(It may be noted that situations such as this are not unprecedented: in the problem of longitudinal diffusion in turbulent pipe flow,for example, the eddy diffusion coefficient is inversely proportional to the molecular diffusivity.)
E. THEFORMSOF clij
AND
fiijk
IN
AXISYMMETRIC TURBULENCE
Suppose now that the turbulence (or random wave field) is not invariant under all rotations, but only under those about the direction defined by the unit vector e. This situation may well arise when the fluid is in a state of mean rotation Lz and when the turbulence is in consequence influenced by Coriolis forces [a situation recently realized and studied in the laboratory by Ibbetson and Tritton (197511;in this case, e = Lz/n and the spectrum tensor (Dij(k,a)is axisymmetric about the direction e. The most general form that (Dij can take in such circumstances,consistent with Hermitian symmetry and the incompressibility condition ki(Dij(k,w ) = 0, is
where
i44 + k245 + p k 4 ,
(5.41)
= 0.
-
Here, 41,. . ., $6 are functions of k, k e = kp, and w, 41,..., 44being real, and 45 and 4 6 complex. The terms involving 41and &z (which are scalar functions) are reflectionally symmetric, while those involving (b3, . .., 4 6 (pseudoscalar functions) are not. The energy spectrum function E(k, p, w ) and helicity spectrum function F(k, p, o)are given by
+ p2)42, ~ ( kp,, w ) = - 4 n i k 2 k , ~ k l m ~=k8nk4{4, l + $(p2 - 1) Im
E(k, p, w ) = 2nk2(Dii= 4nk441 + k(l
(5.42) 46}.
(5.43)
The situation is evidently much more complex than in the isotropic case!
Generation of Magnetic Fields by Fluid Motion
151
The pseudotensors aij and B i j k may still, however, be calculated from formulas (5.27) and (5.28), and symmetry considerations imply that these must now take the form aij = a16 , Bijk
+ a2eiej,
(5.44)
+ 8 2 E j k m e m e i + 8 3 E k i m e m e j + 84Eijmemek + B s d j k e i + f l 6 6 k i e j + B7dijek+ Bseiejek
= BlEijk
(5.45)
where al, a 2 , B5, B6, B7, and BS (being pseudoscalars) may be expressed as weighted integrals of the functions 43, . . ., 4 6 and their derivatives with respect to k and k * e, while PI, B2, /I3, and B4may be expressed similarly in terms of bl and 4 ~The ~ details . of the calculation are not ofgreat interest; it is enough to note that in general &, b2, &, and B4 have the same order of magnitude, as do B s , &, P I , and p8 . The relation (5.3) between 8 and Bo (consideringonly the first two terms) then reduces to the form
8 = a,B0 + a2(e B,)B, - B1 V A B, - B2 e * (V A B,)e - B4(e V)(e A B,) + & V(B, * e) + B,(e * V)B,
.
+ &(e
V)(B, e) + B,(e * V)(B, e)e. (5.46) A
-
The a effect, represented by the first two terms, is now anisotropic but has not suffered any fundamental modification. A nonisotropic a effect of this form still gives rise to amplification of field modes proportional to exp iK * x provided (Moffatt, 1970a) a:K2
+ aIa2(K2- (K
*
e)’)
> A4K4.
(5.47)
Similarly, the eddy diffusivity effect is now anisotropic: the terms involving /I2, f13, and f14 (which occur even in reflectionally symmetric conditions) are
all (presumably) of a dissipative nature like the term involving Bl. The term involving 8 6 is of little interest (unless B6 is allowed to vary slowly as it would in an inhomogeneous field of turbulence) since only the curl of 8 contributes to (5.1). Similarly the term involving B7 may be replaced by - B 7 e A (V A B,) (the difference being irrotational). This effect was discovered by Radler (1969a) and was expressed in the form ylQ A J,, where yl = -B7po/R, and is now generally known as the Q A J effect. The coefficient yl is a pure scalar, and it is tempting to conclude (as did Radler) that the Q A J effect can therefore occur when the random motion is reflectionally symmetric; this conclusion does not, however, appear to be justified, since if @,(k, w) is reflectionally symmetric, then B7 (and so yl) certainly vanishes. This point is of more than academic interest, because it means that if the R A J effect is present, then in general an anisotropic a effect
152
H. K. Mofatt
will be present also. Incorporation of the R A J effect in a model and simultaneous exclusion of the a effect (Radler, 1969b; P. H. Roberts, 1972, Sect. 6) is therefore perhaps somewhat artificial. Nevertheless, it must be said that these models do show that the R A J effect, in conjunction with differential rotation can provide dynamo action, the reason being that the electromotive force y,R A J, tends to generate toroidal current from poloidal and vice versa; the effect therefore provides a possible means whereby the dynamo cycle may be closed. The term involving pawas likewise expressed by Radler in the form (5.48) V)(BO ’ R P , and he argued that if the rotation R, which gives rise to all the nonaxisymmetric contributions to ail and Bilk, is weak, then the term (5.48)being cubic in R should be negligible. This conclusion must also be questioned; as mentioned above, the coefficients /3, and pa are in general of the same order of magnitude, and it would therefore seem inconsistent to include Radler’s R A J effect in any model without including also the term of (5.46) involving Be. Y2W
F. DYNAMO EQUATIONS FOR AXISYMMETRIC MEANFIELDS INCLUDINGMEANFLOWEFFECTS When there is a mean velocity U(x) in addition to the random fluctuations u(x, t), it is no longer realistic to think in terms ofhomogeneous turbulence;
the statistical properties of the turbulence, and in particular the pseudotensors a i l , Bilk, will now themselves be functions of position varying on the same scale as the scale of variation of U(x). The equation for the mean field, which we shall now simply denote B, becomes
aB/dt=VA(UAB)+VI\g+1V2B,
(5.49)
where d is still given by (5.3), in which for the moment we simply assume that we know the form of aU(x)and &,!(x). When B, U, and d are axisymmetric, it is convenient to separate the toroidal and poloidal ingredients of (5.49). If
B=Bi,+B,,,
U=U4+Up,
(5SO)
where U = so(s, z) in cylindrical polars (s, 4, z), then the toroidal ingredient of (5.49) [cf. (3.8), but note that we have now dropped the subscript 4 from B,] is
aB/& + Up VB = s(Bp * V)w
+ (V A &), + l ( V z - s-’)B.
(5.51)
Generation of Magnetic Fields by Fluid Motion
The poloidal ingredient of (5.49) may be “uncurled,” so that if Bp = V the equation for A [cf. (4.1), but with A = x/s] is aAlat
+ up
VA = 8,
+ A(V? - S-
2 ) ~ .
153 A
(A&) (5.52)
Note that the 4, component of electric field E is given by E, = -dA/at; there can be no electrostatic contribution under axisymmetric conditions. Equations (5.51) and (5.52) make it clear that the meridional velocity Up merely has a redistributive effect with regard to both fields A and B ; in the former case, this leads to expulsion of poloidal flux by the mechanism of Section II1,B if 1 is small, although now of course this flux may be regenerated by the 8, term. The term s($ V)o represents generation of toroidal field from poloidal field by the differential rotation mechanism (Section 111,D)This . effectnow acts in conjunction with the source term (V A &), ,and dynamos are of two types, depending on which of these terms is of dominant importance. In the simplest situation in which d = aB - ~ V A B ,
(5.53)
the j3 term leads, as we have seen, to an augmentation of the molecular diffusivity 1, the effective diffusivity being simply 1, = /3 + 1.The a term gives a contribution V A (aBp)in (5.51). The relative magnitude of the two production terms in (5.51) is then given by Is$
- Vwl/(V
A
(.$)I
= O(LZ4)/dl,),
(5.54)
where wb and a. are typical values of 1 Vw I and 1 a 1, respectively (such estimates can of course break down locally). If L2wb Q a 0 , then the differential rotation effect is negligible; the a effectis then solely responsible both for the generation of poloidal field from toroidal and vice versa; dynamos that operate in this way are described as a2 dynamos. The very simple dynamo described at the end of Section V,A is essentially an a2 dynamo, depending as it does on a two-fold operation of the a effect. If L2wb >> ao, on the other hand, then differential rotation is responsible for the generation of toroidal field from poloidal, whereas the a effect, through the term aB in (5.52) regenerates poloidal from toroidal field; a dynamo operating in this way is described as an aw dynamo (avo would perhaps be more accurate, since it is only the gradient of w that is relevant here). The equations for the 010dynamo in the form
+ up VB = - v w + - s - z ) ~ , aA/at + up VA = aB + l ( v z - s - ~ ) A , aB/at
S B ~
rz(v2
(5.55) (5.56)
were obtained originally by Parker (1955a). In this form, the sole effect of the
154
H. K. Moffatt
background random motions is the appearance of the term aB in (5.56), and Parker derived this on the basis of his “random cyclonic events” model (see also Parker, 1970b, 197la,b,c,d,e,f, 1975; Lerche, 197la,b). Equations having the same structure were derived by systematic perturbation procedures by Braginskii (1964a,b), whose approach will be described in the following section ;this latter approach leads, moreover, to the appropriate determination of a in terms of properties of the background fluctuating (nonaxisymmetric) motions. The approach that we have adopted in this section (following Steenbeck, Krause, and Radler) seems the most general and the most easily comprehended-and ease of comprehension is of crucial importance when attention is turned to the far more difficult dynamic aspects of the problem. It is, of course, reassuring that the various approaches, from rather different standpoints, do converge at the same destination [in the form of Eqs. (5.55) and (5.56)], and this lends confidence to the extensive analytical and numerical studies of these equations that have been carried out, some of which will be described in Section VII. VI. Braginskii’s Theory of Nearly Axisymmetric Fields
A. LAGRANGIAN TRANSFORMATION OF THE INDUCTION EQUATION An approach to the dynamo problem that is in some respects complementary to that described in Section V was developed by Braginskii (196hb) and has since been elucidated and extended by Soward (1972).The approach is based on the idea that, although an axisymmetric field cannot be maintained by axisymmetric motions, weak departures from axisymmetry in both velocity and magnetic fields may be sufficient when 1 is small to provide the means of regeneration of the field against ohmic decay. In describing the essence of this theory we follow the approach advocated by Soward. The starting point is a property of invariance of the induction equation in the frozen field (A = 0) limit. For reasons that emerge, it is helpful to modify the notation slightly: let s(%,t ) , a(ji t) represent field and velocity at (2,t), and consider a 1-1 continuous mapping 2 + x(ji t ) satisfying the incom11 be unity. Then if we pressibility condition that the determinant J(dx,/d%, define new fields U , ( q t ) = dx,/df Bk(& t ) = 8, dxk/dzi, the invariance property is that the equation
&/dt = 6 A (ii A 8)
+
&i
dxk/d%i,
(6.1)
Generation of Magnetic Fields by Fluid Motion
155
transforms into aB/at = V A (U A B).
(6.3) B(x, t) is (physically) the field that would result from fs(ji t) under the frozen-field distortion 3 + x. Suppose now that we subject the full induction equation
aB/at - P A(ii A fs) = nP28
(6.4) to the same transformation. We obtain, after some elaborate manipulation of the right-hand side, aB/dt - VA(UAB)= V h 8 + nv2B,
(6.5)
where
8,= a,Bj
+ cikpppjaBk/aXj,
(6.6)
and
Note that aij (a pseudotensor) and Bpj (a tensor) are both quadratic in the displacement x - f. The similarity between (6.5), (6.6) and (5.1), (5.3) is immediately striking; note, however, that in the present context, the term V A 8 is wholly diffusive in origin. Paradoxically, although diffusion is responsible for the natural tendency of a field to decay, it can also be of crucial importance in creating the electromotive force that can counteract this decay.
B. NEARLY AXISYMMETRICSYSTEMS Throughout this section, we shall use angular brackets ( ) to denote an average over the azimuth angle 6 in cylindrical polar coordinates (s, 4, z). Thus for any scalar $(s, 4, z),
With this convention, we may talk of the mean toroidal field (f4)i4 and the mean poloidal field (f,)i, + (f,)i,, which are by definition axisymmetric.
H. K. Moffatt
156
If the velocity field ii(x, t) is nearly axisymmetric, then we may express it in the form
a = (a)
+ EU’,
E
< 1.
(6.10)
A magnetic field P(x, t ) convected and distorted by such a velocity field must exhibit at least a similar degree ofasymmetry, and it is consistent to suppose that
P = (P) + E b .
(6.11)
When E = 0, the field (P)cannot survive, by Cowling’s theorem. Braginskii’s model is based on the expectation that the mean electromotive force E’(u’ A b) may compensate the erosive effects of the term AV’(S) in the averaged induction equation. For this to be possible it is necessary that A be not greater than O(E’);we therefore put A = A. E’ and keep lofixed as E -+ 0. Expansion in powers of E may then be expressed equivalently as expansion in powers of R , ‘I’ where R, = U oL/A,where U o and L are scales chracterThis was the procedure adopted by istic of the mean velocity field (i). Braginskii. The mean velocity may be expressed as the sum of toroidal and poloidal parts:
(ii)
=
U(s, z)i,
+ Up(s,z).
(6.12)
The toroidal part generates toroidal field from poloidal field by differential rotation (Section II1,D) and this is certainly conducive to dynamo action (although, of course, not sufficient in itself). The poloidal part U&, z) tends to redistribute the mean toroidal field; it also tends to wind up the mean poloidal field and to expel it (Section II1,B) from regions of closed streamlines of Up (i.e., from the whole fluid region for an enclosed flow) if a magnetic Reynolds number based on a typical value of I UpI is large. This latter process is certainly not conducive to dynamo action, and the only way to control it is to suppose that I Up / U o is at most O(Rk1), so that
I
UoL/l= R , ,
]Up
= O(1).
(6.13)
The dominant ingredient of the mean velocity field is then the toroidal ingredient Ui, whose typical order of magnitude is U o. To emphasize this scaling, we rewrite (6.12) in the form
(a)
= ~ ( s z)i, ,
+ &’uP(s,
z).
(6.14)
Note that, in proceeding in this way, attention is automatically restricted to velocity fields that have some a priori chance of success as potential dynamos.
Generation of Magnetic Fields by Fluid Motion
157
The mean field (P) may likewise be expressed as the sum of toroidal and poloidal parts :
(P) = B(s, z)i4 + B,(s, z).
(6.15)
The dominant differential rotation Ui4 generates Bi4 from Bp and it may be anticipated (Section II1,D) that B = O(R,) I Bp I, or equivalently I Bp I = O(E’)B.Hence we may also rewrite (6.15) in the form
(P) = ~ ( s z)i4 , + E’L+(s,
z).
(6.16)
It is, of course, implicit in the notation that B, Ib 1, and I b, I are all of the same order of magnitude as E --* 0; similarly for U, Iu’ 1, and Iup I. C. NEARLY RECTILINEAR FLOWS; EFFECTIVE FIELDS It is mathematically simpler to focus attention on the Cartesian analog of the situation described in Section VI,B, in which (s, 4, z) are replaced by ( x , y, z), and ( ) in consequence now indicates an average with respect to the y variable. The additional difficulties of dealing with the cylindrical coordinate system are purely geometrical and need not concern us here. The fact that the fluctuating fields of (6.10) and (6.11) are O ( E )suggests that it may be possible to “accommodate” them through the use ofa transformation function
I=x
+ q(x, t),
(q) = 0,
v - 7 = 0.
(6.17)
If this is possible, then the related fields u and B will have the simple form (the time dependence being for the moment understood)
+ &’ueP(x,z) + O(c3), B = B(x, z)i, + EZbcp(x,z) + O(c3), u = ~ ( xz)i, ,
(6.18) (6.19)
where the effective fields uep and beP are related in some way (to be determined) to the fields up and bp . Let us first obtain the relationship between the fields bepand b, . First, by expanding B,(x) in Taylor series about the point 3 and using Bi(W) = (dik E a1i/dxk)Bk(x),we obtain
+
+
o(E3).
(6.20)
158
H. K. Moflatr
Comparison with (6.11) shows that
b = V A ( ~ A B=) B aq/ay - (q * V)Bi,
(6.21)
(to leading order), and the mean poloidal ingredient of (6.20) gives
(6.22)
-
where q, = q - (q iy)iy is the meridional projection of q. In terms of the vector potentials a, a, defined by b, = V
A
(ai,),
be, = V
(a, i,,),
(6.23)
dqu/ay)y.
(6.24)
A
this result takes the simpler form a = a,
- wB,
w = +(q,
A
The relationship between u,, and up is similarly obtained with the sole difference that it is uk - ax,/at (rather than uk) that appears in the transformation relationship (6.1). This leads to the expression [analogous to (6.21)] = (a/at
+ u a/ayh- (11 - V)ui,,
(6.25)
and the result [analogous to 6.22)]
up = ucp
+ +v A (4,
A
(slat + u aiayh,).
(6.26)
D. DYNAMO EQUATIONS FOR NEARLY RECTILINEAR FLOWS Equations for the evolution of B(x, z, t) and beP(x,z, t) may now be obtained by averaging (6.5) with respect to y. The y component (i.e., the “ toroidal” ingredient) of the resulting equation is aB/at
+ A,,
VB = E2bcp
- vu + (v
A
+
+ o(E41,
B,)~ IV~B
(6.27)
where
(6.28) Since I = O(E’)and aij and /Ipjare both O(E’),A, is evidently O(c4),so that to leading order (6.27) becomes aB/at
+ E2Uep .VB =
&2b,,
- vu + I ,
E ~ v ~ B .
(6.29)
In this equation the effect of departures from exact rectilinearity are wholly absorbed through the introduction of the effective fields uePand be, . Note that the time scale of evolution of the B field is O ( E - ~ L / U ~ ) .
Generation of Magnetic Fields by Fluid Motion
159
The x and z components of the averaged equation (6.5) may be “uncurled ” to give c2 aa,/dt
+ e4uep - Va, = &oy + E2AV2ae+ O(E’),
(6.30)
and here it is clear that the term &‘oy is of the same order as the other terms in the equation. Moreover, the dominant contribution to goy[from (6.2811 is evidently given by g o y = A
+ O(E6),
(6.31)
= 2c2(q3,m q l , m2)9
(6.32)
where, from (6.7) and (6.17), ( ~ 2 2 )=
mql,m2)
where a subscript after a comma indicates differentiation, and (6.30) now becomes aa,/at
+ E ’ U , ~ - Va, = 1TB + 1V2ae,
(6.33)
where r = Equations (6.29) and (6.33) (with the correspondences E ’ U , ~+ Up, &’a,+ A, E2bep +Bp, A(aZ2)-+a) are the Cartesian analogs of the a o dynamo Eqs. (5.55) and (5.56). The effective value of a, given by (6.34) where q is related to the velocity field by (6.25), is noteworthy. The frozenfield displacement merely gives a configuration of u’and b that is potentially able to provide an a effect; it is the diffusive slip of b relative to u’ that realizes this potential and provides an a proportional to A (cf. Section V,C). The structure of (6.34) is elucidated by considering a particular displacement of the form q(x, t ) = qo(cos ky, 0,sin ky),
(6.35)
for which V A q = kq (so that the displacement has positive helicity) and for which (6.34) reduces to a = -Ae2k3q;
.
(6.36)
So again, as in the random wave approach of Section V,C, positive helicity is associated with a negative value for a. More generally, as pointed out by Braginskii (1964a), a displacement of the form q = $)(x, z ) cos ky + q(’)(x, z ) sin ky
(6.37)
will provide a nonzero value for a [given by (6.34)only if the vectors q(‘) and
H.K. Moflatt
160
q(') are linearly independent]. Similarly the parameter w given by (6.24) has
the form W = - k(q(') A ll")),
(6.38)
when q is given by (6.37), and this also is nonzero only when q(') and q(s)are linearly independent vectors. If terms of order higher than E* are retained in (6.29) and (6.33), then new effects are to be expected. These effects turn out, however, to be very much what might be anticipated from the background of mean field electrodynamics as expounded in Section V. To order E', the form of Eqs. (6.29) and (6.33) is in fact unchanged (Tough, 1967; Tough and Gibson, 1969), but the expressions for w and r are modified; in particular, r takes the form To + c r y , where now To is the zeroth-order expression (1/~').with a given by (6.34), and rlinvolves the mean of an expression cubic in q-compare the form of expansion (5.24), which may also be regarded as an expansion in powers of the amplitude of the fluctuating velocity. At order E~ (Soward, 1972) effects analogous to the nonisotropic a and /? effects of Section V,E also appear.
E. COMMENTS ON THE GENERAL APPROACHOF S o W A R D t Soward's approach undoubtedly provides the most natural framework for the treatment of nearly axisymmetric systems in the limit 1+ 0. However, Soward claims that his approach also provides the natural framework even when departures from axisymmetry are not small [i.e., when the transformation function %(x, t ) is not restricted by an assumption of the form (6.17)]. This claim, although plausible, requires some qualification. The reason is that the approach will work, as Soward acknowledges, only for velocity fields ii(ji t) for which a transformation % = %(x,t) exists such that the related field u(x, t) given by (6.la) is predominantly azimuthal, in the sense that u(x, t ) =
q s , 2, t)b + O(R,1).
(6.39)
Now it can easily be shown that the streamlines of the field a(%, t) map into streamlines of u(x, t), which according to (6.39) are nearly the circles C o : s = so, z = z , ; more precisely the topological measure analogous to (2.18) is
t Soward (1972).
Generation of Magnetic Fields by Fluid Motion
161
where u = V A g and u n = 0 on the surface of and for a field of the form (6.39), K = O(R; ’). This means that unless the value of K for the field a(%, t ) is O(R; ’) (for every volume P with surface 3 on which n ii = 0), the transformation function %(x, t ) simply does not exist. In simpler terms, if the streamlines of ii(%) are linked, they cannot be unlinked by a continuous mapping f - r x . In general, a field having both toroidal and poloidal ingredients will have streamlines that are both knotted and linked (Moffatt, 1969) and so the restriction is in fact a severe one. The criticism is not so serious when weak departures from axisymmetry are considered. In this situation Soward (1971) has proved that, given a velocity field of the form
ii = U(s, z)i@+ EU’(S, 4, z),
(6.41)
there exists an axisymmetric V(s, z) such that ii + E’V has streamlines that are closed at the O ( E )level [i.e., if a streamline is followed as 4 increases by 27c, the separation of the endpoints is o(E)].It seems likely that this same choice of V [which is identifiab1e.withueP - up in (6.26)] also guarantees that these same closed streamlines must be unlinked, although a convincing proof of this has yet to be given. A mapping % + x that maps the streamlines of 4 E’V onto the circles Co then exists, and the method can proceed with the necessary adjustments at order E’ in the meridional velocity represented by ueP= up V. Unfortunately there is no analog of Soward’s closed streamline theorem when E is not small; hence the limitation in the class of velocity fields to which the method may be applied when large departures from axisymmetry are considered.
+
+
F. COMPARISON BETWEEN THE TWO-SCALE AND NEARLY AXISYMMETRIC APPROACHES The many points of contact between Braginskii’s theory and the theory of Section V are too numerous to be merely fortuitous and lead to the view that the two approaches are in fact different aspects of one and the same theory. What is essential in both is that it should be possible to define an average ( ). The existence of two scales, I and L, in Section V, lead to the conclusion that d = (u A b) depends only on the local values of B, and its space derivatives and of the statistical properties of the velocity field. In Braginskii’s theory, the two scales are also present in disguised form, the small scale being simply the distance that a fluid particle migrates from its mean position during its passage around the axis of symmetry. The same considerations as in mean field electrodynamics lead inexorably to the conclusion that
162
H . K. Moflai
B must depend only on local values of B(s, z), U(s, z) (and possibly their derivatives), and the mean properties of the fluctuating motion. The crucial conclusion of both theories is that a contribution to B of the form a(B) can arise; and since a is a pseudoscalar in either theory, this effect necessarily requires a lack of reflectional symmetry in the motions that give rise to it; i.e., an observer would be able to distinguish (locally) either a right- or a left-handed character in the motions if the resulting value of a is nonzero. The distinction between uep and g (and between beP and b,,) must be regarded as merely of academic interest until a complete dynamic theory accounting for the structure both of the up field and of second-order mean quantities [like (6.34)] is available; in any case, ifq vanishes at the boundary S of the region V of conducting fluid, then bp = bePon S,and so the matching conditions with the external field are unaffected by the distinction between b, and bep. One is therefore bound to ask what advantage Braginskii's theory has over what appears to be the simpler and more illuminating approach of Section V. The answer is simply that it provides the means whereby the pseudotensors af, and /Ii,! may be derived in the important situation when the mean flow is predominantly toroidal and axisymmetric, the fluctuating velocities are weak, and U,L/A is large. The theory may therefore be recognized as falling within the general framework of the mean field electrodynamics of Section V, but at the same time as occupying a particularly difficult and thorny corner of that subject. It is nevertheless necessary to Dasp the thorns, since the most plausible models for both Earth and Sun do invoke strong toroidal velocity fields and one can hardly simply ignore the influence that this may have on ai, and particularly when dynamical effects are subsequently considered. Perhaps a more obvious attractive feature of Braginskii's approach is the very degree to which it places emphasis on the nonaxisymmetric ingredient ~b of magnetic field. The Earth's field is of course nonaxisymmetric relative to the axis of rotation, the axis of the magnetic dipole moment being inclined at a slowly varying angle I) (at present of order 10") from the rotation axis. A perturbation field having a dependence exp i(4 - of) on azimuth 4 about the rotation axis (Braginskii, 1964c) gives a field with external potential of the form
(6.42)
i.e., a dipole whose moment rotates in the equatorial plane with angular velocity o.In conjunction with the axial dipole, there is here the basis of one (though certainly not the only) possible explanation for the tilt 1(1, and the manner in which the direction of the dipole moment vector drifts relative to the rotation vector.
Generation of Magnetic Fields by Fluid Motion
163
VII. Analytical and Numerical Solutions of the Dynamo Equations
A. THEa' DYNAMO WITH a CONSTANT We have already discussed in Section V,A the simplest exponentially growing solutions of the equation dB/dt = V A ( U A B )+ VA(aB) + 1,V2B,
(74
in the case when U = 0, a = const, and the fluid is of infinite extent, namely the force-free mode for which V A B = KB, for which the growth rate is given by (5.9). Now that we have estimates of a and 1, (= 1+ /3), one further comment on this solution is called for. The growth rate w is clearly maximal when K = K, = ~121,.When R, = u, 111 -g 1 (in the notation of Section V), a lui/1 and A, 2, so that
-
-
IK,
-
R i 4 1,
-
(74
and so the initial assumption that the scale L of the mean field ( K, ') be large compared with the scale 1 of the random u field is amply satisfied. At the other extreme, when R,,, 9 1, we have seen that there is some uncertainty in estimates of a and /3 owing to uncertainty regarding convergence of the integral (5.39), and the unbounded behavior of /3 as given by (5.40). If we assume that a is independent of 1 in the limit 1-+ 0, then on dimensional grounds, a u, in this limit, and so IK, lu, 12,. If 2, is also independent of A in the limit 2 + 0 (as maintained, e.g., by Parker, 1971b) then again on dimensional grounds 1, u, 1. This would lead to the conclusion that IK, = O(l), i.e., that the medium is unstable to magnetic disturbances on scales right down to the scale 1 of the turbulence itself. Of course, if Kl = O(1), then the two-scale approach ceases to be valid; nevertheless the increase in growth rate w with decreasing scale K-' calls for considerable delicacy in the interpretation of results in this limit. This particular difficulty is removed if, as argued at the end of Section V,D, /Iis in fact large compared with lu, when R, B 1. The inadequacy of the traditional estimate /3 lu, in the solar context has been emphasized by Piddington (1972a); however, Piddington's main concern is with the process by which small-scale field fluctuations merge and annihilate each other, and the relevant eddy diffusivity for this process is not necessarily the same as the eddy diffusivity that operates on the global mean field. In this context, see also Parker (1972, 1973). The a' dynamo in a spherical geometry was treated by Krause and Steenbeck (1967).This requires solution of (7.1) (again with U = 0, a = const) in a
-
-
-
-
H.K. Moflatt
164
sphere of radius R, and with matching to a current-free field for r > R. The solutions in this case cannot be force-free, since as mentioned in Section II,D such fields do not exist in a finite geometry in the absence of external sources; the solutions nevertheless undoubtedly have a helical structure. Steady solutions are possible only if the parameter aR/A, takes one of a discrete set of eigenvalues; the lowest of these is 4.49 and the corresponding solution (in r R) for A and B has the form
-=
A = C{k-1r-1’2J,l,(kr) - 3 ~ J ~ , ~ ( k rsin ) } 6, I
i
B = Cr- 1/2J,,2(kr)sin 6,
- -
(7.3)
=-
which matches to a dipole field for r R. If aR/A, D 1 (as would, for example, be the case if a uo, A, uo 1, and 1 6 R) then a wide range of modes become unstable, and the most unstable is not that having the simplest structure (7.3) but rather a high harmonic having a much smaller length scale (as suggested by the results for an infinite geometry).
B. a2 DYNAMOS WITH ANTISYMMETRIC a The assumption of constant a is unrealistic in a rotating body such as the Earth or the Sun in which Coriolis forces are responsible for the generation of helicity. We referred in the introductory section to Parker’s (1955a) concept of rising, twisting blobs of fluid in the convection zone of the Sun surrounded by connected regions of falling fluid. The rising blob tends to entrain fluid from the sides and conservation of angular momentum then leads to spin-up or positive helicity in the northern hemisphere, and similarly to negative helicity in the southern hemisphere. Correspondingly, a would in this picture be negative in the northern hemisphere and positive in the southern. [It is relevant to note that Steenbeck et ai. (1966) came to precisely the opposite conclusion in arguing that a blob that rises through several scale heights will expand and therefore “spin down” in the northern hemisphere, generating negative helicity there. Viscous entrainment and compressibility effects are in competition here and the true sign of a-which is of crucial importance for a o dynamos discussed below-is presumably determined by which of these effect dominates.] The simplest reasonable assumption for a ( x ) on this type of physical picture is a(x) = a , f ( r / R )
cos 6.
(7.4)
Unfortunately, even with this simplest choice, the eigenvalue problem
Vh(aB) + A,V2B = 0
(7.5)
Generation of Magnetic Fields by Fluid Motion
165
(with the matching conditions to a current-free field on r = R) is not amenable to simple analysis, and the problem must be solved numerically (Steenbeck and Krause, 1966, 1969b; P. H. Roberts, 1972), using series expansions for B and truncating after a few terms. Roberts found that six terms were in fact sufficient to give 0.1 % accuracy in the determination of eigenvalues. The most striking feature of the numerical results is that the eigenvalues for fields of dipole and quadrupole symmetry are almost indistinguishable, i.e., these fields can be excited with almost equal ease ! The reason for this was given by Steenbeck and Krause (1969b): the a effect given by (7.4) operates most effectively near the poles, where cos 8 = 4 1, and the toroidal current distribution (which gives rise to the poloidal field) is correspondingly concentrated in rings near the poles. The mutual inductance between these current rings is small, and so one of the current rings can be reversed without greatly affecting conditions near the other. This operation transforms a field of dipole symmetry into one of quadrupole symmetry or vice versa.
IN a o DYNAMOS C. LOCALBEHAVIOR
The Cartesian analog of (5.55) and (5.56) [cf. (6.29) and (6.33)] is
+ up V A = aB + N 2 A , .vu + W B , aB/at + up - V B =
dA/&
(7.6)
*
(7.7) where the variables are functions only of x and z. It is illuminating (Parker, 1955a) to consider the local modes of behavior of solutions of these equations in regions where up, a, and VU may all be regarded as constant. In interpreting these solutions in the solar context, the axis Ox is south, Oy is east, and Oz is vertically upward, relative to some origin 0 in the convective zone in, say, the northern hemisphere. The equations admit solutions proportional to exp(ot iK x) (where K is a real two-dimensional wave vector in the x-z plane), the dispersion relation being
+ . +
(o LK’ + iu, * K)’ = - i a ( K h V U ) , , = 2iy,
(7.8 )
say. If y < 0, the roots of (7.8) are given by o = -1K2 - iup K
k lyI1’’(l - i),
(7-9)
and there is an exponentially growing solution of K satisfies C X J K A V,IU > R2K4.
(7.10)
.
This growing mode has the phase factor exp i(K * x - 9 K t - 1 y Ill’t). If up = 0 and U = U(z) only, then this result implies that if a dU/az < 0 then a
166
H.K. Mofatt
growing mode with space dependence exp(iKx) can propagate in the positive x direction, i.e, toward the equatorial plane. (Alternatively, we could regard o as real and K as complex, in which case the growth of the mode is spatial rather than temporal.) Conversely, if a dU/dz > 0, then the growing mode propagates away from the equatorial plane. The sign of the product of a and the vertical shear is therefore of crucial importance in determining the character of these local solutions, and this carries over also to properties of global solutions of (7.6) and (7.7) in a spherical geometry (Roberts, 1972). Note that when u,, = 0, the above solutions are necessarily oscillatory; the phase velocity can, however, be reduced to zero if up K = - 17 I”’, and we then have an exponentially growing field whose spatial structure remains constant. This property again carries over to global aw dynamos (Braginskii, 1964b; P. H. Roberts, 1972). Suitable choice of meridional circulation can convert a situation in which an oscillatory dynamo (as described in Section II,A) is preferred into one in which a steady dynamo is preferred, i.e., excited at a lower value of 1 D 1 [see (7.12)]. If this meridional circulation is too strong, however, the dynamo fails altogether, presumably because of the consequent expulsion of poloidal flux from the region where the differential rotation would otherwise be generating toroidal field from it. Since differential rotation and meridional circulation are dynamically coupled, the question of whether a given system will exhibit steady or oscillatory dynamo behavior inevitably demands consideration of the governing dynamical equations.
-
D. GLOBAL BEHAVIOROF aw DYNAMOS When a is given by (7.4) and o is assumed to vary only radially, say (7.11)
the crucial dimensionless parameter is what Parker called the dynamo number (7.12)
Generally, when f and g are reasonably smooth overlapping functions and up = 0, (5.55) and (5.56) yield (numerically)only oscillatory solutions (P. H. Roberts, 1972), a solution of dipole symmetry being more easily excited than one ofquadrupole symmetry when D < 0. In such oscillatory solutions, there is a tendency during a period of the oscillation for toroidal field to propagate from polar regions toward the equatorial plane (consistent with the discussion of Section VI1,C); it is this type of behavior that lends credence to the picture relating sunspot formation to the behavior of the underlying toroidal field, as outlined in the introduction. Indeed, many authors (e.g., Steenbeck
Generation of Magnetic Fields by Fluid Motion
167
and Krause, 1969a; Leighton, 1969; Roberts and Stix, 1972)have succeeded, on the basis of such solutions, in constructing butterfly diagrams, which bear at least a plausible resemblance to those based on observation (Maunder, 1922; see, e.g., Kiepenheuer, 1953). Such diagrams depict the appearance of sunspots as a function of latitude and time and are extracted from solutions of the dynamo equations by plotting in the 0-t plane the “isotors” B(r, 8,t ) = const, for some fixed value of r, on the view (Parker, 1955b) that sunspots form by eruption when the toroidal field below the surface exceeds some critical value. When the functions f and g are nonoverlapping, the field most easily excited can be steady rather than oscillating. The case f ( p ) = S(P - pi), g ( P ) = S ( P - PZ), pi # p 2 (7.13) has been studied by Dienzer et al. (1974), who conclude that the steady fieId is favored when I pi - p 2 I is sufficiently large. As an alternative to the eigenvalue approach, Jepps (1975)has carried out a step-by-step numerical integration of (5.55) and (5.56) again for particular choices of the function f and g and for varying values of D. Numerical experimentation leads to determination of the critical value of D at which there exist solutions varying periodically in time without long-term growth or decay. An advantage of this approach is that dynamical effects can be readily incorporated. In particular, Jepps investigated the effect of making a a decreasing function of the local value of B (see Section VII1,A); he found that the sinusoidal time behavior of the poloidal field was distorted by this sort of nonlinear effect toward a “spiked” rather than a “flattened” wave form. Finally, we note that Parker (1971b) has proposed that Eqs. (7.6) and (7.7) with up = 0 may also provide a correct description of the process of generation of the galactic magnetic field (a view that is strongly contested by Piddington, 1972b). With origin 0 at an arbitrary point of the plane of symmetry of the galactic disk (of thickness 2z0), Oz normal to the plane, Ox radial, and with 9
u = G ~ , a=(
a0
-ao
9
,
o
-zo < z < 0,
(7.14)
Parker obtains the dispersion relation for nonoscillatory modes proportional to exp(wt + ikx): for large positive values of the dynamo number, which in this case is D = aGz;/A2, (7.15) and for modes in which B is symmetric about z = 0, this takes the form w
-
- (AD/kz;) exp( -fD’/3) cos
Oil3,
(7.16)
H. K. Moffatt
168
indicating exponential growth if
(4n- 1 ) <~ $D'l3 < (4n + l ) ~ ,
(7.17)
where n is any large positive integer. This is a surprising result since it indicates that for large values of D, a relatively small change (positive or negative) in L) can transform a situation that permits exponential growth of a magnetic field mode into one in which the same mode decays. Of course, it may be that oscillatory modes are more easily excited (Parker, 1971e),as is the case for the spherical geometry; but if not, and if Parker's model is applicable, then the nature of the relation (7.16) between w and D would suggest that the galactic field depends for its existence on a precarious interaction between effects of horizontal shear and cyclonic turbulence of rather critically defined intensity.
VIII. Dynamic Effects and Self-Equilibration A.
WAVES INFLUENCED BY CORIOLIS FORCES, AND ASSOCIATED DYNAMO ACTION
A suitable starting point for the consideration of dynamic effects is an analysis of the waves that can propagate in a rotating incompressible fluid permeated by a locally uniform magnetic field B,. In this context, it is convenient to use the local Alfven velocity V = (p, p)- '/'B0 as the measure of the field strength. The linearized equations governing velocity perturbations u about a state of rest and magnetic perturbations b = (p, p)l/'v are then aulat 2 n A u = -VP v vv + vv2b (8.1)
+
+ aviat = v - vu + A V ~ V ,
(8.2)
v * u = v * v = 0,
(8.3)
where
P =p/p
-t
(v + V)'/2p
- (aAX)',
(8.4) and v is the kinematic viscosity of the fluid. These equations (first considered by Lehnert, 1954) admit solutions of the form (u, v, P) = (a, C,
a) exp i(k
x - at),
(8.5) and (8.2) gives the important relation between C and i already implicit in Section V , C : C = - ( w + iLk2)-'(k V)U. (8.6)
.
Generation of Magnetic Fields by Fluid Motion
169
Equation (8.1) then gives
+ m ~ i=i -ikP,
-iou
(8.7)
where Q
=w
+ ivk’
-
- (w + ilk2)-’(V k ) 2 .
(8.8) By crossing (8.7) with the vector k and then repeating this operation, we obtain two equations from which u and k A ii may be eliminated, giving the dispersion relation CJ
= i2(k
. a)/k,
(8.9)
and the corresponding relation i k A i = _+ki
(8.10)
between the (complex) components of Q. This latter relation implies that such a wave is circularly polarized [cf. thmpatial structure (6.35)] and that its helicity is (u
- 0 )=
f+k1812.
(8.11)
If, therefore, for any reason waves corresponding to, say, the upper sign are present in greater proportion than those with the lower sign, then the net mean helicity will be positive, and the medium will be unstable to the growth of large-scale field fluctuations, by the a2 effect discussed in Section V,A. If now we imagine that the field V itself varies on a large length scale, then large-scale Fourier components of force-free structure [like (2.25) but with 1 K I much smaller than typical values of I k I in the spectrum of u] will grow exponentially. What then ultimately limits the growth of this type of field structure? To see this, it is only necessary to consider the detailed form of the integral (5.27) for aij, which is, of course, still applicable. As V grows locally in strength, the dispersion relation w = w(k) as given by (8.8) and (8.9) is progressively modified in such a way that all components of ai, (which is severely anisotropic in this situation) tend to decrease. Field intensification ceases when these components are reduced to such a level that a effect generation is just compensated by ohmic dissipation. If there is no source of energy for the small-scale motion, then both velocity and magnetic fields must ultimately die away through both ohmic and viscous effects; on the other hand, if random body forces are invoked as a means of maintaining a random wave field, then a steady magnetohydrodynamic equilibrium may be anticipated. This type of process has been analyzed in detail (Moffatt, 1970b, 1972; Soward, 1975). Of particular interest is the level to which the magnetic energy density M = tP’ can grow in this situation (the overbar
H. K . Mofatt
170
indicating an average over the large wavelength 2n/K of the V field). The result obtained (Moffatt, 1972) in the case when waves of typical frequency are maintained by a random force distribution is that in the ultimate steady state the ratio of M to the kinetic energy density E in the random wave field is
M/E
-
(Q/~~)”(v/~)”’(L/I),
(8.12)
where 1 is as usual the scale of the u field and L the scale of the V field. [Viscosity enters into this calculation because, under random forcing, there are some modes (those with k * V = 0) that are unaffected by the magnetic field, and it is only viscosity that can limit the growth of such modes when excited at their resonant frequencies.] Clearly M can be large compared with E if L/l is sufficiently large: there is no question ofequipartition ofenergy in a dissipative system of this kind. If we neglect dissipative effects in the dispersion relation (8.8) and (8.9) (i.e., put 1 = v = 0), then when I V * k 1 4 4(R * k)2/k2,the two roots of the resulting quadratic for w are approximately
-
wfz 2R cos 8 + (232 cos e ) - ’ ( V k)’,
w, z -(2Q cos 8)(V
- k)’
(8.13)
where 8 is the angle betweeli S& dnd k.The root ofcorresponds to an inertial wave slightly modified by the Lorentz force. The root o,( I w, 1 4 lofI) corresponds to a very slow wave (relative to the time scale R-I) in which inertia effects are almost negligible. Indeed, the root o,may be most simply obtained from (8.1)-(8.3) by first dropping the inertia term au/dt. The resulting force balance between Coriolis, Lorentz, and pressure forces is known as magnetostrophic balance [a term introduced by Malkus (1959), although not quite with this same meaning], and we shall therefore describe the waves corresponding to the root o,as magnetostrophic waves. [Hide and Acheson (1973) in their review of the hydromagnetics of rotating fluids use the term “hydromagnetic-inertial” waves, but insofar as inertia plays no part (except via the Coriolis force), the term is perhaps misleading.]
B. MAGNETOSTROPHIC FLOWAND THE TAYLOR CONSTRAINT There is good reason to believe (Hide and Roberts, 1961) that the global force balance in the core of the Earth is approximately magnetostrophic, i.e., that to a good approximation,
2pflAU= -Vp+ JAB,
(8.14)
Generation of Magnetic Fields by Fluid Motion
171
where for the moment we use capital letters U, J, B to emphasize that we have large-scale fields in mind. Suppose that the fluid is contained within a sphere S, center 0, with U * n = 0 on S, and let C(so)be the cylinder s = so, where (s, 4, z ) are cylindrical polar coordinates with Oz parallel to Q. Then integration of the 4 component of (8.14) over C(s,) (Taylor, 1963) gives
JJ
(JAB), d 4 dz = 0,
(8.15)
C(S0)
the integral being over that part of C(s,) contained within S. Under conditions of magnetostrophic balance, therefore, the magnetic field B and associated current J = p;lV A B must satisfy the Taylor constraint (8.15). The result is unaffected by the inclusion of buoyancy forces in (8.14), which of course have no 4 component. We shall refer to important implications of this constraint in Section V1II.D.
C. EXCITATION OF MAGNETOSTROPHIC (MAC) WAVES BY UNSTABLE STRATIFICATION One of the great difficulties in making firm progress on the dynamics of the Earth’s interior is the absence of direct knowledge concerning the dominant source of energy for core motions. One possibility, however, is that at least some part of the liquid core is unstably stratified (either thermally or through coexistence of mixed ingredients of different densities-Braginskii, 1964d);the question of thermal stratification has been discussed by Higgins and Kennedy (1971)and Kennedy and Higgins (1973) on the basis of existing knowledge concerning the melting point temperature of iron under high pressure, and the conclusion of the latter study is that the inner one-third of the liquid core may have a temperature distribution that allows “adiabatic” convection, whereas the outer two-thirds is (in this sense) stably stratified. Whatever the actual situation, the following idealized problem (Braginskii, 1964d, 1967, 1970; Eltayeb, 1972, 1975; Roberts and Stewartson, 1974, 1975) provides a basis for careful analysis of the effect of unstable stratification. Suppose that fluid is contained between planes z = 12, on which the temperature 8 is do f pzo, respectively. Suppose further that Q = (0, 0, Q) and V = (0, r! 0) (other directions also for Q and V are considered in the papers of Eltayeb, 1972, 1975); then, on the Boussinesq approximation, Eq. (8.1) is modified only by the inclusion of a buoyancy force age on the right-hand side, where a is the coefficient of thermal expansion, and d satisfies the linearized heat conduction equation
aelat + pu,
= KvZe,
(8.16)
H. K. Moflatt
172
with K the thermal diffusivity. Equations (8.2) and (8.3) continue to hold unaltered. These equations admit solutions proportional to exp[i(lx + my) + or], with further u,, u,, u x , and uy proportional to cos nz, and u z , uz, and 0 proportional to sin nz. If viscous effects are neglected and if the planes are assumed perfect conductors of heat and of electric current (conditions that are again varied in the papers by Eltayeb, 1972,1975), then all the boundary conditions are satisfied provided nzo /n is an integer. If au/dt is also dropped from the equations (so that attention is focused on the slow magnetostrophic modes) the following cubic equation determining possible values of o is obtained:
Y2(w+ lk2)2(w+ ~
k + (o ~ +) ~
k - X~ ( w) + L k 2 ) = 0,
(8.17)
where
k2 = l2 + m2 + n2,
X = afig(m2
+ I2)/m2k2V2,
Y = 2&/Vzm2k. (8.18)
In the nondissipative limit 3, = K = 0 to which Braginskii ( 1 9 6 4 , 1967) restricted attention, the roots of (8.17) become o = 0 and w = +(X -
(8.19)
1)1’2/Y,
indicating instability (a2> 0) whenever k = (I, m, n) is such that X > 1. In the geophysical context, as in previous sections, we interpret the y direction as east (or azimuth), so that m (like n) can take only discrete values. The most unstable mode is that for which m takes its.smallest nonzero value m , (corresponding to exp i$ dependence on azimuth), and as pointed out by Roberts and Stewartson (1974), if ga/3/V2 exceeds mi by any amount, no matter how small, an infinite number of modes (corresponding to large values of 1) become unstable, since as 1 -+ co,X apg/V2m:. Of course, as 1 increases, these modes are increasingly affected by diffusion and also by inertia [neglected in obtaining (8.17)]. Weak diffusion plays a more important role in shifting the root w = 0 away from the origin of the complex plane. Naive linearization of (8.17) in the small quantities A, K, and o gives for this root the expression
-
W
= -AkZ(X - q)/(x- I),
4 = K/A.
(820)
If 0 < q < X < 1, then the modes given by (8.19)are stable, but that given by (8.20) is unstable with a slow growth rate determined by the weak diffusion effects. This is a “resistive“ instability in the terminology of Furth et al. (1963); its existence underlines the possible importance of diffusion effects, no matter how weak these may be. It is not clear that the arguments of Higgins and Kennedy (1971) concerning the impossibility of radial convec-
Generation of Magnetic Fields by Fluid Motion
173
tion with no diffusive heat exchange apply with equal force when resistive instabilities of this kind are considered. In general, the three roots of the cubic (8.17) are either all real or one is real and the other two are complex conjugates. In this latter case, if the complex conjugate roots are given by o = o,& iw, with o,> 0, the corresponding disturbances propagate as magnetostrophic waves of increasing amplitude. Roberts and Stewartson (1974) show the region of the plane of the variables q and Q = 2iXt/Vz in which such unstable modes are excited in preference to “steady” modes for which m i= 0. The distinction between overstable modes and steady modes is of potential importance in the context of the main problem of regeneration of the large-scale field. We have seen in earlier sections that lack of reflectional symmetry, as measured by the mean helicity (u V A u) is vital in this context; the mean helicity of the unstable motions of the present problem can be calculated on the basis of the linearized equations, and it is found that (u V A u) (the average being over horizontal planes) is nonzero only if mi # 0, i.e., only if the modes are of the overstable variety. It is possible that the nonlinear effects studied by Roberts and Stewartson (1974,1975),and in particular the inclusion of mean shear in the y direction, may modify this conclusion, but this is something that requires further investigation; in this context, see also the discussion of Roberts and Soward (1972, p. 148). At any rate, the picture conceived by Braginskii (1967) is the following: unstable stratification in the liquid core of the Earth in the presence of the predominantly toroidal field excites magnetostrophic waves that are essentially nonaxisymmetric and that propagate in the azimuthal direction. Braginskii called these waves “MAC waves” (M for magnetic, A for Archimedean, C for Coriolis).These nonaxisymmetric motions provide the a effect that is necessary to generate poloidal field from toroidal field, which in turn is generated from the poloidal field by dominant differential rotation. We have here the possibility of a dynamically consistent dynamo driven by buoyancy forces; to be completely consistent, however, the stability analysis described in this section should include the effects of a mean flow in the y direction with strong vertical shear (i.e., the Cartesian analog of strong differential rotation). Roberts and Stewartson (1975) take an important step in this direction. At the same time, a fully consistent model should include the dynamics of the mean flow itself, an important topic on which we focus attention in the following section. Before leaving the topic of convection-driven dynamo action, however, we should refer to the papers by Childress and Soward (1972), Soward (1974), and Busse (1973), in which different approaches to the determination of a dynamically consistent dynamo are considered. Busse considers a combination of two-dimensional convection rolls, together with superimposed shear
-
174
H. K. Moflatt
flow parallel to the rolls. Childress and Soward consider convection in a strongly rotating system in which the horizontal scale of the convection cells is small [O(n-’’’)] relative to the vertical scale. This permits the use of the two-scale methods described in Section V, and leads (Soward, 1974) to the determination of a dynamo in which the magnetic energy density fluctuates about a weak average value. In these studies involving both magnetic field and rotation, it is well known (Chandrasekhar, 1961) that while rotation and magnetic field are separately stabilizing, the two effects can work against each other in such a way that a flow that is stable under the action of rotation alone becomes unstable when a magnetic field is also introduced, while a flow that is unstable under the action of rotation alone becomes even more unstable with a magnetic field. This consideration led Malkus (1959), on the basis of his appealing conjecture that a convective system with infinitely many degrees of freedom will take advantage of this freedom to maximize the transport of heat in the direction of the imposed temperature gradient, to argue that a magnetic field must grow by dynamo action, if only to release the constraints of rotation, to permit more vigorous convection, and so to increase the transport of heat. Soward’s work should in principle provide a test of whether Malkus’s conjecture is true or false; but this still remains a choice target for future investigation. D. MEANFLOW EQUILIBRATION
A magnetic field that is growing in intensity as a result of the a effect will in general not be force-free and will therefore tend to drive a mean velocity, or to modify any preexisting velocity distribution. This effect has been studied by Childress (1969), Malkus and Proctor (1975),and Proctor (1975). It is supposed in these investigations that background turbulence, present for whatever reason, gives rise to an isotropic a effect, with a a prescribed function of position in a sphere, and that there is initially zero mean velocity. It is further supposed that the level of the a effect is just a little greater than that at which field excitation takes place. The growing field B will then approximate to the axisymmetric eigenfunction corresponding to the lowest eigenvalue for a steady a2 dynamo (Sections VII,A and B). A mean axisymmetric velocity field U(x, t) will then develop according to the equation dU/dt + U VU + 2R A U = -VP + ( p o p ) - ‘(V A B) A B + vV2U, (8.21)
and subject to appropriate boundary conditions on the surface of the sphere. The existence of a mean velocity U then modifies the structure of the field B, which evolves according to the equation dB/dt = V A ( U A B )+ Vh(aB) + 1V2B (8.22)
175
Generation of Magnetic Fields by Fluid Motion
in the fluid region. It is to be expected that this modification in structure will lead to increased ohmic dissipation, and hence to curtailment of growth of magnetic energy. This expectation is borne out by numerical integration of (8.21) and (8.22) (Proctor, 1975) in the geophysically relevant case when a = a, cos 8.This problem is characterized by three dimensionless numbers, E = v/f2R2,
EM = A/nR2,
h = a. RIA,
(8.23)
where R is the sphere radius, and the lowest eigenvalue of the kinematic a2 dynamo problem (with U = 0) is in this case (P. H. Roberts, 1972) h, = 7.65. E represents the ratio of viscous forces to Coriolis forces, and, since AIL is the relevant velocity scale, EM represents the ratio of inertia forces to Coriolis forces. Both E and EM are small in the geophysical context; in the limit E = 0, EM = 0, (8.21) degenerates to (8.14) and the balance of forces is then magnetostrophic. In each of the cases studied, Proctor found that the growth of magnetic energy M ( t ) was arrested by the mean flow effect, the level of equilibrium magnetic energy increasing with h - 6,. Provided E and EM were not too small, M ( t ) settled down to a constant level M(oo), after some damped oscillations about this level. ‘Figure 4 shows the structure of the magnetic
(a)
(b)
(C)
(d)
(e)
FIG.4. Field structures in the Proctor (1975) steady state dynamically equilibrated dynamo; E = 0.01, EM = 0.04, B = 8.0. (a) Poloidal field lines, (b) isotors B = const, (c) lines of constant angular momentum about axis of symmetry, (d) lines of constant vorticity of meridional circulation, (e) streamlines of meridional circulation.
field and the velocity field in the ultimate steady state when E = 0.01, EM = 0.04, and h = 8.0. The poloidal field is shown only for r c R, although it of course extends outside the sphere. There is an indication of a “cylindrical structure” in the patterns of angular momentum and vorticity, which is presumably an indication of the influence of the Taylor constraint (8.15), which is nearly satisfied for these low values of E and EM. At even lower values of E and EM (E = 0.005, EM = 0.0025), Proctor found that the oscillations of M ( t ) about its ultimate mean level did not show any tendency to die out with time. These oscillations of M ( t )are associated with
176
H.K. Moflatt
torsional oscillations about an equilibrium in which the Taylor constraint is satisfied. As pointed out by Roberts and Soward (1972) such torsional oscillations will be damped by Ekman suction effects in boundary layers on the sphere r = R, but when E and EM are very small, this damping is evidently not sufficient to suppress the oscillations. It appears that the steady equilibrium state satisfying the Taylor constraint is unstable or in some other sense unattainable; a related phenomenon was found by Roberts and Stewartson (1975) in the treatment uf nonlinear aspects of the stability problem discussed in Section VII1,C. Neglect of the suppression of the a effect by the growing magnetic field is justified in Proctor’s investigation for small values of h - h, since then the mean flow equilibrium mechanism causes a leveling off of M ( t )at a low level at which the turbulence suppression effect is still negligible. The two effects would have to be considered in conjunction if h - 1 5 were ~ large; but then one would also have the complication that more than one mode of the linearized kinematic problem might be unstable. The model is also open to criticism on the grounds that the reflectionally asymmetric turbulence giving rise to the a effect is simply assumed present, without any consideration of the means whereby it may be maintained. A marriage of the models of Sections VIII,C and D may in the longer term provide something approaching the complete picture as far as maintenance of the Earth’s field is concerned.
ACKNOWLEDGMENTS The writing of this review was completed during my tenure of a Visiting Professorship at the lnstitut de Mtcanique Theorique et Appliqute, Universite Pierre et Marie Curie, Paris, and I wish to express my gratitude to Professor Cabannes and his colleagues for generous support and hospitality during my visit. I am indebted to Dr.R. H. Kraichnan who drew my attention to an erroneous conclusion in the original draft of Section V,C; this led to an important modification of the penultimate paragraph of that section.
REFERENCES
BABCOCK,H. W, and BABCOCK,H. D. (1955). The Sun’s magnetic field. Astrophys. J . 121, 349-366.
BACKUS,G. E. (1958). A class of self-sustaining dissipative spherical dynamos. Ann. Phys. N e w York 4, 372-441. BATCHELOR, G. K . (1950). On the spontaneous magnetic field in a conducting liquid in turbulent motion. Proc. Roy. Soc.,Ser. A 201, 405-416. BATCHELOR,G. K. (1953). “Homogeneous Turbulence.” Cambridge Univ. Press, London and New York.
Generation of Magnetic Fields by Fluid Motion
177
BATCHELOR, G. K. (1956). Steady laminar flow with closed streamlines at large Reynolds numbers. J . Fluid Mech. 1, 177-190. S. I. (1964a). Self excitation of a magnetic field during the motion of a highly BRAGINSKII, conducting fluid. Sou. Phys.-JETP 20,726-735. BRAGINSKII, S. I. (1964b). Theory of the hydromagnetic dynamo. Sou. Phys.-JETP 20, 1462-147 1.
BRAGINSKII, S. I. (1964~).Kinematic models of the Earth’s hydromagnetic dynamo. Geomag. Aeron. 4, 572-583. BRAGINSKII, S . I. (1964d). Magnetohydrodynamics of the Earth’s core. Geomag. Aeron. 4, 698-7 12.
BRAGINSKII, S . I. (1967). Magnetic waves in the Earth’s core. Geomag. Aeron 7, 851-859. BRAGINSKII, S. I. (1970). Oscillation spectrum of the hydromagnetic dynamo of the Earth. Geomag. Aeron. 10, 172-181. BULLARD, E. C. (1968). Reversals of the Earth’s magnetic field. Phil. Trans. Roy. SOC.London, Ser. A 263,481-524. BULLARD, E. C., and GELLMAN, H. (1954). Homogeneous dynamos and terrestrial magnetism. Phil. Trans. Roy. SOC.London, Ser. A 247, 213-278. Buss& F. H. (1973). Generation of magnetic fields by convection. J . Fluid Mech. 57, 529-544. BUSSF, F. H. (1975). A necessary condition for the geodynamo. J . Geophys. Res. 80,278-280. CHANDRASEKHAR, S. (1956). On force-free magnetic fields. Proc. Nat. Acad. Sci. U.S. 42, 1-5. CHANDRASEKHAR, S. (1961). “ Hydrodynamic and Hydromagnetic Stability.” Oxford Univ. Press (Clarendon), London and New York. CHILDRESS,S. (1969). A class of solutions of the magnetohydrodynamic dynamo problem. I n “The Application of Modern Physics to the Earth and Planetary Interiors” (S.K. Runcorn, ed.), pp. 629-648, Wiley, New York. CHILDRESS,S, and SOWARD, A. M.(1972). Convection driven hydromagnetic dynamo. Phys. Rev. Lett. 29, 837-839. COWLING, T. G. (1934). The magnetic field ofsunspots, Mon. Notic. Roy. Astron. SOC.94,39-48. DIENZER, W, VON KUSSEROW, H. U., and STIX,M. (1974). Steady and oscillatory am-dynamos. Astron. Astrophys. 36, 69-78. DILLON,R. F. (1975). Gravity and magnetic field correlation and related geomagnetic topics. Ph.D. Thesis, Cambridge Univ., Cambridge, England. S. (1975). “On Magnetic Dynamo Mechanism of the Planets,” Prep. N9a (124). DOLGINOV, Inst. Terr. Magn., Ionosphere, Radio Wave Propagation, Moscow. S, YEROSHENKO, Y.,and ZHUZGOV,L. (1973). Magnetic field in the very close DOLGINOV, neighbourhood of Mars according to data from the Mars 2 and Mars 3 Spacecraft. J. Geophys. Res. 78, 4779-4786. DROBYSHEVSKI, E. M, and YUFEREV, V. S. (1974). Topological pumping of magnetic flux by three-dimensional convection. J . Fluid Mech. 65, 33-44. ELTAYEB, I. A. (1972). Hydromagnetic convection in a rapidly rotating fluid layer. Proc. Roy. SOC.,Ser. A 326, 229-254. ELTAYEB, I. A. (1975). Overstable hydromagnetic convection in a rotating fluid layer. J . Fluid Mech. 71, 161-179. FURTH,H. P, KILLEEN, Jv and ROSENBLUTH, M. N. (1963). Finite-resitivity instabilities of a sheet pinch. Phys. Fluids 6, 459-484. GAILITIS,A. (1970). Selfexcitation of a magnetic field by a pair of annular vortices. Mag. Gidrod. 6, 19. [Magnetohydrodynamics 6, 14 (1972).] GIBSON,R. D. (1968). The Herzenberg dynamo 11. Quart. J . Mech. Appl. Math. 21, 257-267. GOLITSYN, G. S. (1960). Fluctuations of the magnetic field and current density in a turbulent flow of a weakly conducting fluid. Sou. Phys. Dokl. 5, 536-539.
H. K . Moflatt
178
GUBBINS, D. (1974). Theories of the geomagnetic and solar dynamos. Rec;. Geophys. Space Phys. 12, 137-154.
HERZENBERG, A. (1958). Geomagnetic dynamos. Phil. Trans. Roy. Soc. London, Ser. A 250. 543-585.
HIDE,R. (1974). Jupiter and Saturn. Proc. Roy. SOC.,Ser. A 336,63-84. HIDE, R, and ACHESON, D. J. (1973). Hydromagnetics of rotating fluids. Rep. Progr. Phys. 36, 159-22 1.
HIDE, R, and ROBERTS, P. H. (1961). The origin of the main geomagnetic field. Phys. Chem. Earth 4, 25-98. HIGGINS,G . H, and KENNEDY, G. C. (1971). The adiabatic gradient and the melting point gradient in the core of the Earth. J . Geophys. Res. 76, 1870-1878. A,, and TRITTON, D. J. (1975). Experiments on turbulence in a rotating fluid. J . Fluid IBBETSON. Mech. 68, 639-672. JEPPS, S . A. (1975). Numerical models of hydromagnetic dynamos. J. Fluid Mech. 67,625-646. KENNEDY, G. C, and HIGGINS,G. H. (1973). The coreparadox, J . Ceophys. Res. 78,900-904. KIEPENHEUER, K. 0.(1953). Solar activity. In ‘The Sun” (G. P. Kuiper, ed.), pp. 322-465. Univ. of Chicago Press, Chicago, Illinois. KOVASZNAY, L. S. G. (1960). Plasma turbulence. Rev. Mod. Phys. 32, 815-822. KRAICHNAN, R. H. (1973). Helical turbulence and absolute equilibrium. J. Fluid Mech. 59, 745-752.
KRAICHNAN, R. H, and NAGARAJAN, S. (1967). Growth of turbulent magnetic fields. Phys. Fluids 10, 859-870. KRAUSE,F. (1968). Eine Losung des Dynamo problems auf der Grundlage einer linearen Theorie der magnetohydrodynamischen Turbulenz. Habilitationschrift, Univ. of Jena, Jena. (See Roberts and Stix, 1971.) KRAUSE,F, and ROBERTS, P. H. (1973). Some problems of mean field electrodynamics. Astrophys. J . 181,977-992. KRAUSE,F.. and STEENBECK, M. (1967). Some simple models of magnetic field regeneration by non-mirror-symmetric turbulence. Z . Naturforsch. A 22, 671475. (See Roberts and Stix, 1971.)
LEHNERT, B. (1954). Magnetohydrodynamic waves under the action of the Coriolis force. Astrophys. J . 119, 647-654. LEtGHmx, R. €3. (1969). A magneto-kinematic model of the solar cycle. Astrophys. J . 156, 1-26.
LERCHE, I. (197 la). Kinematic-dynamo theory. Astrophys. J . 166, 627-638. LERCHE, I. (1971b). Kinematic-dynamo theory, 11. Dynamo action in infinite media with isotropic turbulence. Astrophys. J . 166, 639-649. LIEPMANN, H. W. (1952). Aspects of the turbulence problem. Z. Angew. Math. Phys. 3,321-342. LIGHTHILL,M. J. (1959). “Introduction to Fourier Transforms and Generalized Functions.” Cambridge Univ. Press, London and New York. L~RTZ,D. (1968). Impossibility of steady dynamos with certain symmetries. Phys. Fluids 11, 913-915.
F. J, and WILKINSON, I. (1963). Geomagnetic dynamo: A laboratory model. Nafure (London) 198, 1158-1160. LOWS, F. J., and WILKIXSON, I. (1968). Geomagnetic dynamo: An improved laboratory model. Nature (London) 219, 717-718. LUMLEY, J. L. (1962). The mathematical nature of the problem of relating Lagrangian and Eulerian statistical functions in turbulence. In “Mecanique de la Turbulence.” No. 108, pp. 17-26. CNRS, Paris. W. V. R. (1959). Magnetoconvection in a viscous fluid of infinite electrical conductiMALKUS, vity. Asrrophys. J . 130, 259-275. Low-
Generation of Magnetic Fields by Fluid Motion
179
MALKUS, W. V. R.,and PROCTOR, M. R. E. (1975). The macrodynamics of a-effect dynamos in rotating fluids. J. Fluid Mech. 67, 417-444. MAUNDER, E. W. (1922). The Sun and sunspots 1820-1920. Mon. Not. Roy. Astron. SOC. 82, 534-544. H. K. (1961). The amplification of a weak applied magnetic field by turbulence in MOFFATT, fluids of moderate conductivity. J. Fluid Mech. 11, 625-635. MOFFATT,H. K. (1963). Magnetic eddies in an iircumpressible viscous fluid of high electrical conductivity. J. Fluid Mech. 17, 225-239. MOFFATT,H. K. (1969). The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35, 117-129. H. K. (1970a). Turbulent dynamo action at low magnetic Reynolds number. J. Fluid MOFFATT, Mech. 41, 435-452. MOFFATT,H. K. (1970b). Dynamo action associated with random inertial waves in a rotating conducting fluid. J . Fluid Mech. 44,705-719. MOFFATT,H. K. (1972). An approach to a dynamic theory of dynamo action in a rotating conducting fluid. 1.Fluid Mech. 53, 385-399. MOFFATT,H. K., ed. (1973). Report on the NATO Advanced Study Institute on magnetohydrodynamic phenomena in rotating fluids. J , Fluid Mech. 57, 625-649. H. K. (1974). The mean electromotive force generated by turbulence in the limit of MOFFATT, perfect conductivity. J . Fluid Mech. 65, 1-10. K. W., LEPPING,R. P., WHANG,Y. C, and SCHAITEN,K. H. (1974). NESS, N., BEHANNON, Magnetic field observations near Mercury: Preliminary results from Mariner 10. Science 185, 151-160. PARKER, E. N. (1955a). Hydromagnetic dynamo models. Astrophys. J. 122, 293-314. E. N. (1955b). The formation of sunspots from the solar toroidal field. Astrophys. J. PARKER, 121,491-507. PARKER, E. N. (1963). Kinematical hydromagnetic theory and its applications to the low solar photosphere. Astrophys. J. 138, 552-575. PARKER, E. N. (1970a).The origin of Solar magnetic fields. Ann. Rev. Astron. Astrophys. 8, 1-30. E. N. (1970b). The generation of magnetic fields in astrophysical bodies. I. The dynamo PARKER, equations. Astrophys. J. 162, 665-673. E. N. (1971a). The generation of magnetic fields in astrophysical bodies. 11. The PARKER, galactic field. Astrophys. J. 163, 255-278. E. N. (1971b). The generation of magnetic fields in astrophysical bodies. 111. Turbulent PARKER, diffusion of fields and efficient dynamos. Astrophys. J. 163, 279-285. PARKER, E. N. (1971~).The generation of magnetic fields in astrophysical bodies. IV. The Solar and terrestrial dynamos. Astrophys. J. 164, 491-509. PARKER, E. N. (1971d). The generation of magnetic.fields in astrophysical bodies. V. Behaviour at large dynamo numbers. Astrophys. J. 165. 139-146. E. N. (1971e). The generation of magnetic fields in astrophysical bodies. VI. Periodic PARKER, modes of the galactic field. Astrophys. J. 166, 295-300. PARKER, E. N. (1971f). The generation of magnetic fields in astrophysical bodies. VIII. Dynamical considerations Astrophys. J. 168, 239-249. E. N. (1972). Topological dissipation and the small-scale fields in turbulent gases. PARKER, Asfrophys. J. 174, 499-510. PARKER, E. N. (1973). The reconnection rate of magnetic fields. Astrophys. J. 180, 24% PARKER,E. N. (1975). Generation of magnetic fields in astrophysical bodies, X. Magnetic buoyancy and the solar dynamo. Astrophys. J. 198, 205-209. PARKER, R. L. (1966). Reconnexion of lines of force in rotating spheres and cylinders. Proc. Roy. SOC., Ser. A 291, 60-72.
180
H . K. Moflatt
PICHAKHCHI, L. D. (1966). Theory of the hydromagnetic dynamo. Sou. Phys.-JETP 23, 542-543. PIDDINGTON, J. H. (1972a). Solar dynamo theory and the models ofBabcock and Lcighton. Sol. Phys. 22, 3-19. PIDDINGTON, J. H. (1972b). The origin and form of the galactic magnetic field, 1. Parker’s dynamo model. Cosmic Electrodyn. 3, -70. PROCTOR.M. R. E. (1975). Non-linear mean field dynamo models and related topics. Ph.D. Thesis, Cambridge Univ., Cambridge, England. RADLER,K.-H. (1969a). On the electrodynamics of turbulent fluids under the influence of Coriolis forces. Monats. Dr. Akad. R’iss. Berlin 11, 194-201 (see Roberts and Stix, 1971). RADLER,K.-H. (1969b). A new turbulent dynamo, 1. Monatsber. Deut. Akad. Wiss. Berlin 11, 272-279. (See Roberts and Stix, 1971.) ROBERTS,G. 0. (1970). Spatially periodic dynamos. Phil. Trans. Roy. Soc. London, Ser. A 266, 535558. G. 0.(1972). Dynamo action of fluid motions with two-dimensional periodicity. Phil. ROBERTS, Trans. Roy. Soc. London, Ser. A 271, 411-454. ROBERTS, P. H. (1967). “An Introduction to Magnetohydrodynamics.” Longmans, Green, New York. ROBERTS,P. H. (1971). Dynamo theory. I n “Mathematical Problems in the Geophysical Sciences” (W.H.Reid, ed.), pp. 129-206. Amcr. Math. Soc.,Providence Rhode Island. ROBERTS, P. H. (1972). Kinematic dynamo models. Phil. Trans. Roy. Soc. London, Ser. A 272, 663-698. ROBERTS,P. H., and SOWARD,A. M.(1972). Magnetohydrodynamics of the Earth’s core. A n m . Rev. Fluid Mech. 4, 117-154. A. M.(1975). A unified approach to mean k l d electrodynamics. ROBERTS, P. H, and SOWARD, Astron. Nachr. 2%. 49-64. ROBERTS, P. H., and STEWANEON, K. (1974). On finite amplitude convection in a rotating magnetic system. Phil. Trans. Roy. SOC. London, Ser. A 277, 287-315. ROBERTS, P. H, and STEWARTSON, K. (1975). On double roll convection in a rotating magnetic system. J . Fluid Mech. 68, 447-466. P. H, and STIX,M. (1971). “The Turbulent Dynamo: A Translation of a Series of ROBERTS, Papers by F. Krause, K.-H. Radler and M.Steenbcck,” Tech. Note NCAR/TN/lA-60. Nat. Cent. Atmos. Phys, Boulder, Colorado. ROBERTS, P. H., and STIX,M. (1972). a-ElTect dynamos by the Bullard-Gellman formalism. Astron. Astrophys. 18, 45M66. SAFFMAN,P. G. (1960). On the effect of the molecular diffusivity in turbulent diffusion. J . Fluid Mech. 8. 273-283. SAFFMAN, P. G. (1963). On the fine-scale structure ofvector fields convected by a turbulent fluid. J . Fluid Mech. 16, 545-572. S C H L ~ RA, , and BIERMANN, L. (1950). Interstellare Magnetfelder. Z . Naturforsch. A 5, 237-251. SMITH,E., DAVIS,L., JONES,D. E.,COLBURN, D. S., COLEMAN, P. J, DYAL, P, and SONETT, C. P. (1974). Magnetic field of Jupiter and its interaction with the Solar wind. Science 183, 30-306. SOWARD,A. M.(1971). Nearly symmetric advection. J . Math. Phys. ( N . Y . ) 12, 2052-2062. SOWARD, A. M.(1972). A kinematic theory of large magnetic Reynolds number dynamos. Phil. Trans. Roy. Soc. London, Ser. A 272, 431-462. SOWARD,A. M.(1974). A convection-driven dynamo I. The weak field case. Phil. Trans. Roy. SOC. London, Ser. A 275, 61 1-651.
Generation of Magnetic Fields by Fluid Motion
181
SOWARD,A. M. (1975). Random waves and dynamo action. J. Fluid Mech. 69, 145-177. STEENBECK, M., and KRAUSE,F. (1966). The generation of stellar and planetary magnetic fields by turbulent dynamo action. Z. Naturforsch. A. 21,1285-1296. (See Roberts and Stix, 1971.) STEENBECK, M., and KRAUSE, F. (1969a). On the dynamo theory of stellar and planetary magnetic fields, I. A.C. dynamos of solar type. Astron. Nachr. 291.49-84. (See Roberts and Stix, 1971.)
STEENBECK, M., and KRAUSE, F. (1969b). On the dynamo theory of stellar and planetary magnetic fields, 11. D.C. dynamos of planetary type. Astron. Nachr. 291, 271-286. (See Roberts and Stix, 1971.) STEENBECK, M., KRAUSE, F., and R ~ L E RK.-H. , (1966). A calculation of the mean electromotive force in an electrically conducting fluid in turbulent motion, under the influence of Coriolis forces. Z. Naturforsch. A 21, 369-376. (See Roberts and Stix, 1971.) TAYLOR,G. I. (1922). Diffusion by continuous movements. Proc. London Math. SOC.,A 20, 196211. (Collect. pap., Vol. 11, pp. 172-184. Cambridge Univ. Press, London and New York, 1960.) TAYLOR, J. B. (1963). The magnetohydrodynamics of a rotating fluid and the Earth’s dynamo problem. Proc. Roy. SOC.London, Ser. A 274, 274-283. TOUGH, J. G. (1967). Nearly symmetric dynamos. Geophys. J. Roy. Astron. SOC.13, 393-396. TOUGH,J. G., and GIBSON, R. D. (1969). The Braginskii dynamo. In “The Application of Modern Physics to the Earth and Planetary Interiors” (S. K. Runcorn, ed.), pp. 555-569. Wiley, New York. S. I., and ZEL’DOVICH, Y. B. (1972). Origin of magnetic fields in astrophysics. Sou. VAINSHTEIN, Phys.--Usp. 15, 159-172. VENEZIAN, G. (1967). “Magnetohydrodynamics of the Earth’s Magnetic Field,” Rep. No. 85-37, Div. Eng. Appl. Sci., California Inst. Tech., Pasadena, California. WARWICK, J. (1963). Dynamic spectra of Jupiter’s decametric emission. Astrophys. J. 137,41-60. WEISS,N. 0. (1966). The expulsion of magnetic flux by eddies. Proc. Roy. Soc., Ser. A 293, 310-328.
WEBS, N. 0. (1971). The dynamo problem. Quart. J. Roy. Astron. SOC.12, 432-446. WEISS,N. 0.(1974). Dynamo maintenance of magnetic fields in stars. In “Magnetohydrodynamics” (L. Mestel, F. Meyer, and N. 0.Weiss, eds.). Swiss SOC.Astron. Astrophys, Geneva. WOLTJER,L. (1958). A theorem on force-free magnetic fields. Proc. Nat. Acad. Sci. U.S. 44, 489-491.
ZEL’WVICH,Y. B. (1957). The magnetic field in the two-dimensional motion of a conducting turbulent liquid. Sou. Phys.-JEW 4, 460-462.
This Page Intentionally Left Blank
The Theory of Optimal Load Transmission by Flexure GEORGE I . N . ROZVANY AND ROBIN D. HILL Faculty of Engineering Monash University Chyton, Victoria. Australia
I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . I1 . Static-Kinematic Optimality Criteria-Plastic Design . . . . . . . . . A. Basicconcepts . . . . . . . . . . . . . . . . . . . . . . . . B. Unspecified Cost Distribution and a Single Load System . . . . . . C . Unspecified Cost Distribution-Several Alternate Loads . . . . . . D. Unspecified Cost Distribution-Multicomponent Systems . . . . . . E . Partially Prescribed Cost Distribution-Fixed Segmentation . . . . F. Generalized Loads and Reactions of Unspecified Magnitude and Nonzero Cost in Given Location . . . . . . . . . . . . . . . . . . G . Joints (Connections) of Nonzero Cost . . . . . . . . . . . . . . H. Discontinuous Cost Functions . . . . . . . . . . . . . . . . . I . Partially Prescribed Cost Distribution-Optimal Segmentation . . . J . Generalized Reactions and Loads of Unspecified Magnitude and Location . . . . . . . . . . . . . . . . . . . . . . . . . . . K. A Superposition Principle . . . . . . . . . . . . . . . . . . . 111. General Optimality Criteria-Elastic Bsign . . . . . . . . . . . . . A. Basicconcepts . . . . . . . . . . . . . . . . . . . . . . . . B. Strength Design . . . . . . . . . . . . . . . . . . . . . . . . C . Design for Compliance or Deflection Constraints . . . . . . . . . D . Optimal Location of Supports. Hinges, and Segment Boundaries . . . IV . Optimal Flexure Fields-Basic Geometrical Properties . . . . . . . . A. Derivation of Optimal Moment-Curvature Rate Relations . . . . . B. Properties of S-and T-Type Regions . . . . . . . . . . . . . . . C . Properties of R-Type Regions . . . . . . . . . . . . . . . . . . D. Proof of Propositions 1-6 . . . . . . . . . . . . . . . . . . . E . Topographical Properties of Optimal Flexure Fields . . . . . . . . F. Axially Symmetric Supports and Loading . . . . . . . . . . . . . V . Optimal Flexure Fields-Clamped Boundaries . . . . . . . . . . . . A . Review of Literature on Nonaxisymmetric Optimal Flexure Fields . . B. Properties of Solutions for Clamped Boundaries . . . . . . . . . . C . Proof of Properties 1-6 . . . . . . . . . . . . . . . . . . . . . D. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
184 187 187 190 193 194 195 196 197 198 199 199 201 201 201 202 204 205 206 206 209 210 214 218 220 226 226 227 231 239
George I . N.Romany and Robin D . Hill
184
VI. Optimal FlexureFields-Mixed Boundary Conditions . . . . . . A. Branches Containing a and 8 Fields Only . . . . . . . . . . B. Proof of Properties 7-14 . . . . . . . . . . . . . . . . . C. ct and B Fields-Examples , . . . . . . . . . . . . . . . D. y and 6 Fields . . . . . . . . . . . . . . . . . . . . . . E. Branches Containing y and 6 Fields , . . . . . . . . . . . F. Examples of Other Types of Interfield Boundaries . . . . . . VII. Optimal Flexure Fields-Further Developments . . . . . . . . A. The Effect of Combined Bending and Torsion . . . . . . . . B. The Effect of Combined Bending and Shear . . . . . . . . . C. Solutions for Internal Supports . . . . . . . . . . . . . . D. Combined Upward and Downward Loads . . . . . . . . . E. Optimal Grillages for Alternate Loads . . . . . . . . . . . F. Solutions for Other Problems Involving Optimal Flexure Fields G. Published Results Superseded by the Present Theory . . . . . H. The Problem of Free Edges . . . . . . . . . . . . . . . . VIII. Optimal Flexure Fields of Constrained Geometry . . . . . . . . List of Symbols . . . . . . . . . . . . . . . . . . . . . . References.. . , . . . . . , . . . . . . . . . . . . . . .
. . . . . . . . . . .
.
. . 246 . . 246 . . 250 . . 252 . . 272 . . 282 . . 291 . . 292 . . 292 . . 293 . . 294 . . 296 . . 296 . . 296 . . 296 . . 299
. . . . . . . 299 . . . 303 . . . 304
I. Introduction A turning point in the history of structural optimization was marked by the findings of Michell(lW4), who outlined optimality conditions for trusslike continua in which the cross-sectional areas are proportional to the absolute value of the member forces. The class of grillagelike continua to be considered in this study is very similar to Michell structures, as can be seen from the comparison given in Table 1. TABLE 1
A COMPARISON OF OFTIMALTRUSSLIKE AND GRILLAGELIKE CONTINUA Trusslike continua (Michell frames)
Grillagelike continua
Loads are parallel to the middle plane
Loads are normal to the middle plane of the structure Cross-sectional area of beams is proportional to the absolute value of the bending moments in beams Principal curvatures take on a constant value in the direction of nonzero principal moments
of the structure Cross-sectional area of members is proportional to the absolute value of the axial forces in members Principal strains take on a constant value in the direction of nonzero principal forces
Optimal Load Transmission by Flexure
185
A considerable amount of research effort has been expended on Michell's problem over several decades. It is rather remarkable, therefore, that whereas only a relatively few exact solutions are available for Michell's criteria, rigorous analytical solutions for minimum weight grillages can now be obtained for almost any boundary condition and loading. This conclusion is even more surprising if we consider the fact that grillages may have any combination of three types of boundary conditions (clamped, simply supported, or free edges) while the choice of support conditions for Michell frames is more limited. The main problem to be considered herein may be described as follows: Let D be a plane horizontal area (Fig. 1) subjected to a system of vertical loads (e.g., P , and P, in Fig. l), which are to be transmitted to specified
FIG.1. The main optimization problem in this article.
supports (S in Fig. 1) by a finite or infinite set of beams having unspeciJied centroidal axes along D and a continuously variable width but a prescribed constant depth.? The total weight of the beams is to be a minimum within a given design constraint. It has been found that the optimal moment field is the same for any one of the following three criteria: (i) Rigid-plastic grillages: prescribed ultimate load. (ii) Linearly or nonlinearly elastic grillages: prescribed maximum stress. (iii) Linearly elastic grillages: prescribed value of the work of the specified loads on the actual elastic deflections (compliance constraint). The same moment fields minimize the volume of continuously variable reinforcement in fiber-reinforced plates of constant thickness.The foregoing t While Fig. 1 shows a finite number of beams, actual optimal solutions for this problem often consist of an infinite number ofbeams havingan infinitesimal width. This is the reason for using the term grillagelike continuum, which was introduced by Prager (19744.
186
George I . N . Rozvany and Robin D. Hill
problem is also referred to as the “minimum moment volume problem” because it gives a minimum value for the integral of the sum of the absolute values of principal moments, considering the set of all statically admissible moment fields for given boundary conditions and loading. Concise but lucid expositions of the grillage problem have been presented in recent lectures by Prager (1973, 1974a, 1975). The theory discussed has several useful applications, such as beam systems in floors and roofs and reinforced concrete slabs. Actual tests on full-scale optimized slabs have been carried out successfully at Monash University (Rozvany, 1973h). On the other hand, designs based on Michell trusses are somewhat less realistic because they assume the same principal stress value both in tension and in compression. Due to buckling of compression members, this assumption is contrary to actual design practice. Another interesting aspect of grillage design is that the quantitative implications of optimization appear to be more significant than for most other types of structures. To illustrate this point on a simple example, consider the problem in Fig. 2a in which four point loads P are required to be transmitted by beams of constant depth to simple supports along the boundary of a square domain. In the fmt solution shown (Fig. 2b), the total moment area, which is a measure of structural weight, is 6Pa2. The optimal beam layout in Fig. 2c gives a total moment area of 4 P d . Whereas the 50%
tc)
Fio. 2. Comparison of optimal and nonoptimal beam layouts.
Optimal Load Transmission by Flexure
187
difference in this example is significant enough, much greater savings have been indicated for other boundary conditions. The volume of conventional (isotropic) reinforcement in circular footing slabs, for example, has been found to be 540% higher than the optimal reinforcement volume for the same ultimate load (Romany and Adidam, 1971). Unconstrained optimization of constant depth grillages often yields solutions of limited practical value because the complexity of the geometry renders them suitable only for long-span structures or repetitive production. Therefore, the possibility of introducing various simplifying constraints has also been investigated and is reviewed in Section VIII. Although the main topic of this paper is the constant-depth grillage problem introduced in Fig. 1 and discussed in Sections IV-VII, optimal load transmission by flexural systems may take on a variety of other forms depending on the design constraints adopted. Whereas these other flexural problems are not treated at depth in the present study, a comprehensive set of optimality criteria for members in bending is listed without proofs but with references to previous publications in Sections I1 and 111. In some cases, it has been possible to present relatively general optimality criteria in a concise form (Sections QA-F). In considering other problems, however, a general formulation would have been too lengthy, and therefore the criteria are stated in the particular context of beams (Sections II,GJJII). Since a grillage is a plane system of beams, all extremum conditions listed can be used readily in grillage problems, as can be seen from the applications in Sections VII and VIII. All optimality criteria presented are extensions or generalizations of theories of W. Prager and his associates, whose terminology and notation are employed throughout this study. A list of frequently used symbols is given on pages 303 and 304.
11. Static-Kinematic Optirnality Criteria-Plastic Design
A. BASICCONCEPTS Using Prager’s (1974a) unified terminology, the four classes of parameters used in structural theories are generalized loads P(x) = [Pl(x), . . ., P,(x)], generalized displacements p(x) = [pl(x), ..., p,(x)], generalized stresses Q(x) = [Q1(x),. . ., Qm(x)],and generalized strains q(x) = [ql(x), . . ., qm(x)], where x is the spatial coordinate. The corresponding load Piand displacement p i components, or stress Q j and strain qj components, are defined by stating that the specijc external
188
George I . N . Rozuany and Robin D. Hill
work w ( ~of) the loads P on the displacements p is n
and the specijc internal work w ( * )of the stresses Q on the strains q is m
]= 1
In the case of perfectly plastic structural elements, generalized strains and displacements, respectively, are replaced by generalized strain rates and velocities, and the specific work by specijc power. The relation between generalized loads and stresses is given by the static continuity conditions (Prager, 1974a) or, in other words, equilibrium equations, and between displacements and strains by the kinematic continuity conditions or compatibility equations. In addition, strain-stress relations express the relationship between generalized strains and stresses. For example, the static and kinematic continuity conditions and the generalized strain-stress relations for linearly elastic beams are
- u , =~K , ~
M J E I = K, 12.3 ) where M ( x ) is the bending moment, P ( x ) the transverse load, u ( x ) the deflection, K ( X ) the curvature, E Young’s modulus, I the moment of inertia, and x the longitudinal coordinate. Subscripts following a comma indicate differentiation with respect to the variable shown. In addition to the foregoing three relations, the generalized displacements and stresses, respectively, must obey kinematic and static constraints (boundary conditions). If a beam, for example, is clamped at x = 0, is unsupported, and is not subjected to a point load at x = L, then the kinematic and static constraints become M s x x= - P ,
M ( L ) = M,,(L) = 0.
u(0) = U , , ( O ) = 0,
(2.4)
Generalized displacements and strains satisfying kinematic continuity conditions and kinematic constraints are called kinematically admissible, and generalized loads and stresses satisfying static continuity conditions and static constraints are called statically admissible. Thequantity Q which is to be minimized subject to design constraints, will be termed cost. It can be expressed in the form (Prager and Shield, 1967)
R
=
i,$ dx,
where D is the structural domain and $(x) the specific cost or cost per unit
Optimal Load Transmission by Flexure
189
volume, area, or length. Although a more general formulation is possible, it will be assumed that the specific cost at a given point x depends on the generalized stress only:
$(XI
(2.6)
= $[Q(x)l.
Equation (2.6) is referred to as the specijc cost function. Restricting $(Q) to piecewise differentiable functions, let the stress space Em have rectangular coordinates Q , , . . ., Q , and let II/ = be differentiable on the interior of the stress regimes S, ( I = 1, . .., r), where Em = S1 u u S, and Int Sin Int S, = 0 for i # j . Then the generalized gradient operator 9 is defined as
$(a)
grad $ = ($,Ql,..., $,am),
for Q* E Int S,, (2.7) for Q** E Bd S,, (2.8)
n
,€I
with
c v,
= 1,
v, 2 0,
lot.
where grad, $(a**) =
lim
grad $(Q:*),
with Q:*
E
Int S, , (2.10)
Qp+*-+Q*+
and the symbols Bd and Int, respectively, refer to the boundary and interior of a set. The meaning of (2.8)-(2.10) is explained graphically in Fig. 3, where the stress vector Q has only two components, Q1and Q 2 .
FIG.3. Graphical representation of the generalized gradient operator.
$(a)
On the interior of the stress regimes S1 and S2 (Fig. 3a), is differentiable. The slope of $(Q), however, is discontinuous across the common boundary AB of the stress regimes S1 and S2and thus the gradient of $( ) at Q** is not defined in conventional vector calculus. If, however, the gradient is expressed as a limiting value of the gradients for a sequence of points Q:*
190
George I . N . Romany and Robin D . Hill
(Fig. 3b) that (i) are contained in the interior of one of the stress regimes S, ( I = 1 or 2), but (ii) converge to a point Q** that is contained in the common boundary AB (Q:* -+ Q**),then a different gradient grad, $(Q**)will be associated with each stress regime at Q** (see vectors in thick line at Q** in Fig. 3a). The summation and multipliers v, in (2.8) with the restriction in (2.9) also admit any convex combination of the limiting gradients grad, $(Q**)( I E I ) . The latter are represented by vectors in thinner lines at Q** in Fig. 3a. The generalized gradient operator Y defined in (2.7)-(2.10)has been found very useful in expressing extremum principles in structural optimization. Optimality conditions for various classes of design problems with a particular emphasis on flexural systems are summarized in this section. For proofs, the reader will be referred to earlier papers. In optimal design problems, static and kinematic continuity conditions are usually specified, but the generalized strain-stress relations are not known because the cross-sectional geometry is not prescribed fully prior to design. However, the extremum condition (min 0)can usually be converted into a special strain-stress relation representing a condition for optimality and thus the problem of optimal design, in effect, is transformed into a problem of ana1ysis.t Optimality criteria expressed in the form of a strainstress relation will be referred to as static-kinemric optimalicy conditions.
B. UNSPECIFIED COSTDISTRIBUTION AND A SINGLE LOADSYSTEM The idea of stating optimality criteria in the form of a strain-stress relation was introduced by Prager and Shield (1967),who generalized a notion by Marcal and Prager (1964).If there are no constraints imposed on the cost distribution, then the Prager-Shield optimality condition for optimal plastic design can be expressed as (2.11) where q and Q, respectively, are kinematically and statically admissible.The same condition has been confirmed by variational methods (Charrett and Romany, 1972a). For the particular case of a plastic rectangular beam having a given constant depth and variable unspecified width and designed for bending stresses only, the specific cost function representing the cross-sectional area becomes
*
= klMI,
t This method was employed extensively by Martin (1971).
(2.12)
Optimal Load Transmission by Flexure
191
where k is a known constant and M the bending moment. For this case, (2.7)-(2.11) give the optimality condition IC = k sgn M , for M # 0, (2.13)
1.
for M = 0,
Ik,
(2.14)
in which K is the curvature defined in (2.3). Relation (2.13) was first stated by Heyman (1959). If the specific cost of the beam depends on both the bending moment M and shear force V in the form $ = k J M I +kilJ'l,
(2.15)
then the optimality condition becomes (Rozvany, 1973a) IC = k
sgn M,
for M f0,
y = kl sgn V, for
1x1 Ik,
for M = 0 ,
I y I Ik l , for
V#O,
(2.16)
V=O,
where y is the shear strain such that the beam deflection u ( x ) is given by x
u(x)=
[ [
x
I c d x d x + [ x y d x + A x + B,
Jo Jo
(2.17)
JO
where A and B are constants of integration. The type of specific cost function shown in (2.15) can be used for trusses, for example, because in the latter the weight of chords and web, respectively, is controlled by the bending moment and shear force. The Prager-Shield criterion (2.11) is a necessary and sufficient condition for global optimality if the specific cost function $(Q) is convex and the equilibrium equations are linear. It can also be employed usefully, as a necessary condition, if the specific cost function is nonconvex (Rozvany, 1973b; Rozvany and Adidam, 1972a). In the particular case of homogeneous speciJic cost functions of order p, which may be defined by the statement $(kQ) = kp$(Q),
for all k > 0,
(2.18)
the minimum cost may be calculated from any one of the following three expressions (Rozvany, 1973~): (2.19) (2.20) (2.21)
192
George 1. N . Rozvany and Robin D . Hill
in which Q, q, P,and p are statically and kinematically admissible and satisfy (2.11). Naturally, (2.19) is valid for any convex specific cost function. For homogeneous specific cost functions, the optimality criterion (2.11) also implies the condition q Q/$ = d'/$ = const
on D,
(2.22)
which is termed the uniform energy dissipation principle and was obtained originally by Drucker and Shield (1956, 1957). Homogeneous specific cost functions of order p with (0 < p < co)may also be defined by making use of the operator Y:
Q . S[$(Q)I= MQ).
(2.23)
For homogeneous specific cost functions of order one, the cost gradient Y[$(Q)] depends only on the direction of the stress vector Q but not on its magnitude :
Y[$(kQ)l= S[lCI(Q)],
for k > 0.
(2.24)
The locus of Y[$(Q)] for such functions is termed the cost gradient surface. Upper and lower bounds, respectively, on nmin may be obtained by evaluating the following quantities (Rozvany and Adidam, 1972d): =
jD *(Q? dx,
(2.25)
1
(2.26)
fi = P * pk dx, D
in which Q" is statically admissible, and pk is kinematically admissible and is associated with strain rates qk, which represent points on or inside the cost gradient surface. The validity of (2.26) is restricted to convex homogeneous specific cost functions of order one. For the same class of functions, the Prager-Shield (Prager and Shield, 1967) optimality condition (2.11) may be given another physical interpretation, namely, the optimal stress and strain rate fields (Q,,,, q,,,) are identical with stress and strain fields for a structure of ideal locking material (Prager, 1957), having the same static and kinematic continuity conditions and constraintst as the plastic system to be optimized and a locking surface that is identical with the cost gradient surface of the latter. This means that the Prager-Shield condition (2.11) transforms the problem of optimal design of perfectly plastic systems to the problem of analysis of structures made out of a locking material.
t In earlier terminology: the same boundary conditions, loading, equilibrium, and compatibility equations.
Optimal Load Transmission by Flexure
193
If absolute limits [t,hmin(x), $max(x)]are imposed on the specific cost, then (2.11) must be supplemented by the following conditions (Prager, 1974a): q=
,I*[$(a)], " I *[$(Q)]
for $(Q)< $min for $(Q)= $mi"
$(a)=
for
$ma,
(2.27)
3
9
3
0 I v I 1,
(2.28)
1 I v I a*
(2.29)
C. UNSPECIFIED COSTDISTRIBUTION-SEVERAL ALTERNATE LOADS Let a structure be subject to alternative loads Ph(X) (h = 1, .. ., t) and let the corresponding velocities, stresses, and strain rates be ph(x), Qh(x),and qh(x), respectively. At a point x E D,each load condition Ph(X)gives a different cost requirement $(Qh).The design value $ of the cost is then given by
(2.30) For alternate loads, the Prager-Shield optimality condition (2.11) changes to (Rozvany and Adidam, 1972d; Charrett and Rozvany, 1972a)
(2.31)
qh = &y[$(Qh)l, Ah
> 0,
only if
$(Oh) =
$, h = 1, ..., t,
(2.32) (2.33)
where qh and Qh( h = 1, .. ., t), respectively, are statically and kinematically is given by the following two admissible. Then the minimum cost amin expressions :
(2.34) (2.35) Upper and lower bounds on the minimum cost are furnished by max
=
"=s where
q h
D
$(a;)dx,
(2.36)
h
iPh'PkdX,
(2.37)
D h=l
is statically admissible and pk is kinematically admissible and is
George I . N . Rozvany and Robin D . Hill
194
associated with strain rates
(2.38) in which Y* may define any point on or inside the cost gradient surface. Equation (2.37) is restricted to convex homogeneous specific cost functions of order one. Minimum-weight plastic design of beams and frames for alternate loads are discussed by Mayeda and Prager (1967) and Prager (1967),respectively. If a structure having the domain D is subjected to an infinite number of loading conditions such that each load system is associated with a vector 6 E B (where B is the set of all admissible values of t),then (2.33) and (2.32) change to
A(% 6)d6 = 1,
(2.39)
for all x E D,
1.4
A(x,
S) > 0,
only if $(x) = max c
$[Qk 6)l = $[Q(x,513,
(2.40)
where $(x) is the design value of the specific cost at x and Q(x, 6)is the state of stress at x equilibrating the load associated with vector 6. It is important to note that 6 may define not only the location but any other property of a load system. For example, the alternate load systems may consist of a line load of length Land intensity P with LP = const, and 6 may then take the form
(2.41)
6 = ( y , z, 8,L),
in which y and z are the coordinates of the centroid of the line load, 8 defines its orientation, and L its length. Problems involving moving loads were discussed by Save and Prager (1963), Save and Shield (1966), and Lamblin and Save (1971).
D. UNSPECIFIED COSTDISTRIBUTION-MULTICOMPONENT SYSTEMS The design value $ of the specific cost may consist of several components $k ( k = 1, . . ., v ) such that v
(2.42) (2.43) where
Qh
equilibrates the hth load condition and
I(/k(Qh)
are termed cost
Optimal Load Transmission by Flexure
195
component functions. The foregoing problem differs from the one discussed in Section II,C, because the cost components $k are independent from each other at any point x and thus they may be governed by different load conditions. An example of a multicomponent system is an axisymmetric fiber-reinforced plate having top and bottom reinforcing fibers in radial and circumferential directions. Extension of the Prager-Shield optimality condition (2.11) to this problem gives (Charrett and Rozvany, 1972a) U
qh
=
h = 1,
&g[$k(Qh)]t k= 1
t,
(2.4) (2.45)
qk= +k(Qh),k = 1, ..., u,
A,, 2 0 and Ak,, > 0 only if
h = 1, ..., t,
(2.46)
where qh and Qhare, respectively, kinematically and statically admissible. An upper bound on the minimum cost SZmin is given by (2.47)
where Q; (h = 1, . . ., t ) are statically admissible. A lower bound on the minimum cost for convex homogeneous cost component functions of order one is furnished by (2.37),in which pf:is kinematically admiskble and is associated with a strain rate field v
(2.48)
with
c &, t
= 1,
k
= 1,
. . ., 0, &,2 0,
(2.49)
h= 1
where 9; is a vector representing a point on or inside the cost gradient surface for t+kk(Q). E. PARTIALLY PRESCRIBED COST DISTRIBUTION-FIXED SEGMENTATION Let the domain D be divided into segments D, (g = 1, . . ., w ) and let the design cost value $ ( x ) be prescribed in the form
196
George I . N . Rozvany and Robin D . Hill
where Ag ( g = 1, . . ., w ) are unknown constants and [,(x) are given functions termed shape functions. Then the Prager-Shield optimality condition takes on the following modified form (Rozvany, 1973~): q h = &Y[d'(Q,)],
h = 1,
.
.) t,
(2.51) (2.52)
1, > 0
only if $ ( x ) = IC/(Qh).
(2.53)
Special cases of the foregoing problem were discussed by Foulkes (1954), Prager (1971% 1974a), and Sheu and Prager (1969). If absolute limits (Agmin,Agma.Jare imposed on the multiplier A,, then for Ag = Agminand Ag = Agmsxthe equality sign in (2.52) changes to 2 and ,I respectively. An upper bound on Rminmay be obtained by finding statically admissible stress fields Q; (h = 1, . . ., t ) that satisfy (2.50) and then integrating $ over D. For homogeneous specific cost functions of order one, a lower bound on Rminmay be established by evaluating (2.37) such that pk are kinematically admissible and are associated with strain rates satisfying (2.51) and (2.52), in which Y[$(Qh)] is replaced by a vector Y* representing any arbitrary point on or inside the cost gradient surface for $(Oh).
F. GENERALIZED LOADSAND REACTIONSOF UNSPECIFIED MAGNITUDE AND NONZERO COSTIN GIVENLOCATION Considering a perfectly plastic structure having the domain D and the specific cost function $(Q),let the generalized reactions (consisting of forces and couples) or loads R acting on a given subdomain S E D be associated with the cost function $(R). Then the optimization problem becomes
(2.54) For an unspecified distribution of $ ( x ) , the extended version of the Prager-Shield optimality condition for the above problem becomes (Rozvany, 1974) 4 x 1 = %[$(Q)I, on D. (2.55) P(X)
= 9[4(R)],
on
s.
(2.56)
In the case of several alternative loads, (2.31)-(2.33) replace (2.55), and (2.56) must also be changed in a similar manner. For a partially prescribed be distribution of $, (2.51)-(2.53) replace (2.55). An upper bound on Rmincan
Optimal Load Transmission by Flexure
197
obtained from any statically admissible system of stresses and reactions on the basis of (2.54). For homogeneous cost functions ($ and 4 ) of order one, a lower bound on Qmi, is given by (2.57) if pk is kinematically admissible with
qk =8*($), on D,
(2.58)
pk = Y*(4),
(2.59)
on S,
where Q*($) and %*(4),respectively, denote any vector representing a point on or inside the cost gradient surfaces for $ and 4. For convex homogeneous specific cost functions of order p, Omi,,may also be calculated from
where p satisfies (2.55) and (2.56). G. JOINTS (CONNECTIONS) OF NONZERO COST
The optimality criteria presented in this section are discussed in the particular context of one-dimensional continua (e.g., beams, frames, rings) but they can be extended readily to multidimensional systems. Assuming that connections of nonzero cost occur at x = x i (i = 1, . . ., z ) and have an infinitesimal length but nonzero cost, the total cost for the system becomes (2.61) where qi( ) is the cost function for the joint at x = x i . For the foregoing problem, the Prager-Shield condition (2.1I) changes to (Rozvany and Mroz, 1975) Q = Q[$(Q)]+
Q { ~ i [ Q ( x i ) ]S(X > - Xi),
(2.62)
i= 1
where 6 is the Dirac delta function (impulse function) and Q and q, respectively, are statically and kinematically admissible. The idea of including the cost of joints in the optimization procedure is due to Prager (1974b) and has also been used by Parkes (1975).
198
George I . N . Romany and Robin D. Hill
(b)
(a)
FIG.4. Optimal velocity fields for joints of nonzero cost.
Considering the connection of a beam, the joint cost function qi( ) may take on the following form: Vi(M
V ) = c l M I +c,
pq,
(2.63)
where c and c1 are known constants, M the bending moment, and V the shear force. Then at x = x i slope discontinuities (relative angular velocities) Oi and velocity discontinuities Ai (Fig. 4) take on the following values (Rozvany and Mroz, 1975):
Oi = c sgn M ( x i ) , lei
1 5 C,
Ai = c1 sgn V(x,),
IAil
IC,,
for M ( x i )# 0,
(2.64)
for M ( x i )= 0,
(2.65)
for
V ( x , )# 0,
(2.66)
for
V ( x , )= 0.
(2.67)
Remark. All optimality criteria discussed in Sections I1,B-G represent necessary and sufficient conditions for a (global) minimum of the cost if the cost functions involved are convex. The criteria discussed in Sections I1,H-L hereinafter are only necessary, even in the case of convex cost functions. Further, the latter criteria will be discussed only in the context of beams under flexure, although their extension to other systems is readily available.
H. DISCONTIN uous COSTFUNCTIONS
+,
Let the specific cost function +(M)for a beam have a discontinuity at M = M i(Fig. 5a). Then the Prager-Shield criterion (2.11) can still be used
M, (a)
DO&O+!
M
(b)
(C)
FIG.5. Problems involving discontinuities in the specific cost.
Optimal Load Transmission by Flexure
199
as a necessary condition if an angular velocity 8 is inserted at the cross sections (x = X) with M = M i . The magnitude of 8 is given by (Rozvany, 1974b)
6=
$i//
V(x)l,
(2.68)
where V is the shear force (Fig. 5b). Modified conditions have been derived for cases when V is discontinuous or M has a local extremum at x = X. I.
PARTIALLY PRESCRIBED
COST DISTRIBUTION-OPTIMAL SEGMENTATION
Considering again a beam with a momentdependent specific cost function $(M), let the cost distribution be prescribed in the form given in (2.50) and let the boundaries of the segments D, (g = 1, . . ., w )be unspecified and to be optimized. Denoting the cost discontinuity at a segment boundary again by A$i (Fig. 5c), (2.51)-(2.53) together with (2.68) will constitute a necessary condition for optimality. This means that in the optimal velocity field u(x), the angular velocities 8 furnished by (2.68) will appear at plastic hinges at the segment boundaries. In relation to optimal segmentation, a proof of (2.68) based on principles of mechanics was obtained by Prager (Prager and Rozvany, 1975) and a variational proof by Rozvany (19744.
J. GENERALIZED REACTIONSAND LOADS OF UNSPECIFIED AND LOCATION MAGNITUDE If the specific cost $(M)is moment dependent and a load or reaction consists of a normal force R1 in an unspecified location x B and has a cost function 4 ( R , , xB),then a necessary condition for the optimal location and magnitude of R , becomes (Rozvany, 1975a)
*;
- $I; + 0,v,
- 0;v; + +,,
= (P,RI~
= 0,
(2.69) (2.70)
where $; and I); are specific costs (Fig. 6), 0;and 0: absolute angular velocities, and V ; and V : shear forces at infinitesimal distances to the left and right of B. If the cost q5( ) does not depend on the location x B of the reaction and there is no cost discontinuity at B, then (2.69) reduces to (Fig. 6)
v,
0;
v;,
= 0;
(2.7 1)
which has also been obtained by Prager (Prager and Rozvany, 1975), who used principles of mechanics.
200
George I . N . Rozvany and Robin D . Hill
(b) FIG.6. Condition for optimal support location.
If, in addition, the cost distribution is unspecified and $(M)is continuous, then there is no slope discontinuity (relative angular velocity) at Band hence (2.71) reduces to WB
= 0,
for R , = 1;'
- V , # 0,
(2.72)
where oB= u ,is ~the absolute angular velocity at B. Equation (2.72) has also been obtained by Mroz (Mroz and Rozvany, 1975). If the reaction R , has a zero cost, then (2.70) changes to u s = 0.
(2.73)
I
If the specific cost is shear dependent in the form $( V ) = k V I and the cost of the load or reaction R , is independent of its location x B , then (2.71) is replaced by (Rozvany, 1975a)
-v,
=
v;.
(2.74)
Considering now an unspecified reaction or load consisting ofa couple R , having a cost function 4 ( R 2 , xB), a necessary condition for the optimal location and magnitude of R , becomes (Rozvany, 1975a)
ll/i - $; + 4.u = = (b.R2'
0 9
(2.75) (2.76)
= 0 and thus (2.75) becomes For location-independent reaction cost, 4,x8
*,
= ?;$
(2.77)
which has also been derived by Mroz (Mroz and Rozvany, 1975). Conditions (2.69)-(2.77) can be extended readily to several alternate load
Optimal Load Transmission by Flexure
20 1
conditions. For example, the generalized form of (2.71) is t
t
(2.78) where the subscript h denotes quantities associated with the hth load condition.
K. A SUPERPOSITION PRINCIPLE
A very useful principle has been introduced by Nagtegaal and Prager (1973), who have shown that an optimal plastic design for two alternative loads P, and P, may be obtained by first deriving optimal solutions for the loads P+ = +(PI + P2) and P- = i(P, - P,) and then superimposing the values of the strength parameters in the solutions for P+ and P-. The same principle was stated in the context of linear programming and trusses by Hemp (1973). 111. General Optimality Criteria-Elastic
Design
A. BASICCONCEPTS In minimizing the cost of an elastic structure, one of the following constraints may be introduced: (i) Strength constraints, where the maximum permitted value of stresses is specified. (ii) Compliance constraints, where the work of generalized loads on the actual elastic displacements is prescribed :
where C is a given constant. (iii) Dejection constraints, where the generalized displacements at a given point or some linear combination of the displacements at a finite or infinite number of points is prescribed. This constraint may be expressed in the form
Jb*P p dx = C, where D* G D, P(x) is a given vector function, and p(x) is the elastic displacement field associated with the loads P(x). P(x) can be regarded as a
202
George 1. N . Rozvany and Robin D . Hill
virtual load system that may contain point loads and distributed loads. For example, a constraint on the deflection or rotation, respectively, at a single point corresponds to a virtual load P(x) consisting of a single point load or couple. Clearly (3.1) is a special case of (3.2). B. STRENGTH DESIGN The field of application of the Prager-Shield condition (2.11) can be extended to optimal elastic strength design if it gives a generalized stress field Q(x) for which the corresponding elastic strains q,,(x) are kinematically admissible. Considering beams obeying Bernoulli’s hypothesis, the foregoing cond ition is, in general, satisfied if the specific cost function JI( ) and stiffness S = E l are related to the moment capacity M in the following form: $ ( M ) = k l IM(”,
S = k , IM12-”.
(3.3)
Then (2.1 1) gives the following optimal plastic curvature rates:
(3.4)
K=klalMP-’sgn M . By (2.3), the corresponding elastic curvatures are K , ~=
MIS = M / k , I M
I P-’
= M
sgn M l k ,
(3.5)
Since (2.1 1) requires K(X) to be kinematically admissible, K , , ( x ) also satisfies the kinematic constraints if the latter consist of prescribed zero displacements at rigid supports, giving 4x1
= 4l(X)7
(3.7)
after integrating (3.6) twice. However, KJX) satisfying (3.6) may be kinematically inadmissible if a given deflection is assigned to a point as in the case of the optimality condition in (2.70). Several problems of practical importance fall into the category defined by (3.3). For various types of rectangular beams, for example, cc in (3.3) takes on the following values:
(i) constant depth but variable width: a = 1; (ii) constant width but variable depth: a = 4; (iii) constant depth/width ratio: a = 4. The first case above is of particular interest to us, because a set of beams of
Optimal Load Transmission b y Flexure
203
(b) M, =PLI8
-Mo=-PL/8
(el
K=-1
K=-1
A
dX)
(0
I
-+
K
FIG.7. Example in which optimal elastic and plastic designs coincide.
prescribed depth constitutes the grillages in the main problem discussed in this study. The foregoing conclusions imply that optimal plastic design of grillages of prescribed constant depth also minimizes their structural weight in elastic strength design. Optimal plastic and elastic strength design may yield the same results even in cases when (3.6) is violated. To demonstrate this point, we shall consider a clamped beam subjected to a central point load P (Fig. 7a). Assuming that the beam has a prescribed constant depth but variable width, which must exceed a specified minimum value, let the specific cost function
204
George I . N . Rozvany and Robin D . Hill
for the beam take the form (Fig. 7b) $(M)=
kJM(,
for M 2 M o = P L / 8 ,
(3.8)
klMoI,
for M s M , ,
(3.9)
where k is a constant and M o is the yield moment of the cross section having the prescribed minimum width. The curvature requirements furnished by (2.11) are shown in Fig. 7c and the optimal statically admissible moment diagram ( M ) together with the corresponding velocity field ( u ) and curvaturerates (K)for k = 1 in Fig. 7d-f. The elastic curvature rates induced by the same moments take on the values shown in Fig. 7g if all cross sections with M 2 M o develop the same permissible stress. In Fig. 7 g ko = M O P 0
9
(3.10)
where So is the elastic stiffness of the cross section having the prescribed minimum width. It will be seen that the curvatures in Fig. 7g are also kinematically admissible, and thus the solution in Fig. 7d is optimal for both plastic design and elastic strength design in spite of the fact that (3.6) is violated. However, a simple calculation shows that the two solutions do not coincide if the point load on the beam in Fig. 7a is replaced by a uniformly distributed load. If (2.11) gives a moment field for which the elastic curvatures would be kinematically inadmissible without discontinuities in the slope, then the solution can still be rendered optimal in elastic strength design if it is stipulated that, in order to restore kinematic admissibility, hinges may be inserted at cross sections where the optimal moment field takes on a zero value.
c. DESIGN FOR COMPLIANCE OR DEFLECTION CONSTRAINTS Let the design constraints for a beam be of the type shown in (3.1) or (3.2) and let the specific cost $ depend on the elastic beam stiffness S of unspecijied distribution in the form $ = $(S), where S is required to exceed a given value S o . Then the optimality condition becomes (Rozvany, 1975a) $,s = c M M / S 2 ,
for S > S o ,
(3.11)
2 cMM/S2,
for S = S o ,
(3.12)
$,s
where c is a constant, M and A, respectively, equilibrate with the loads P and virtual loads P [see (3.2)], and both curvature fields MIS and M / S are
Optimal Load Transmission b y Flexure
205
kinematically admissible.? For the case of $ ( S ) = S, (3.11) and (3.12) were obtained by Prager (1971b). If the stiffnessdistribution is prescribed on segments D1,. . .,D, in the form S = A8&,(x),
on D,, g = 1, ..., w,
(3.13)
then the optimality condition changes to (Rozvany, 1975a) (3.14) For the particular case of $ ( S ) = S and LJx) = const, (3.14) reduces to 1 = (c/A,ZL,)
.6,M A dx,
(3.15)
which was first obtained by Prager (1971b). Equations (3.11) and (3.12) are of particular interest to the main grillage problem discussed herein because for beams of given depth $ ( S ) = kS and for the compliance constraint in (3.1) P = F, M = A,and hence by (3.11) and (3.12) k = M 2 / S 2 = K:, ,
for S 2 0,
(3.16)
k 2 M 2 / S 2 = K:, ,
for S
(3.17)
= 0,
which gives the same solution as the static-kinematic criteria (2.13)and (2.14) for optimal plastic design. Since, in a sense, the compliance constraint prescribes the average stiffness of a structure for its design load, the PragerShield condition (2.11) minimizes the weight of elastic grillages of given depth both for a prescribed strength and for a prescribed average stiffness.
D. OPTIMAL LDCATION
OF
SUPPORTS, HINGES,AND SEGMENT BOUNDARIES
Considering a deflection constraint given in (3.2) and introducing the notation
5 = $(S) + c M A / S ,
(3.18)
the condition for the optimum location X~ of a support with Q hinge becomes (Rozvany, 1975a)
I);
+ we Pi +
V , - I); - w B+ PB - QB+ VB+ = 0,
in which w; and w i are slopes and V ; and
(3.19)
V i are shear forces associated
t If the slope of $ ( S ) is discontinuous for some values of S, then +,s is replaced by 9 [ $ ( S ) ] in (3.11), (3.12). and (3.14).
206
George I . N . Romany and Robin D . Hill
with the real loads P, and the corresponding quantities with overbars are associated with the virtual loads P. The superscripts + and - denote the fact that the quantity is taken at an infinitesimal distance to the right or left of the cross section B. A specific cost discontinuity may be due to a segment boundary at B or to a discontinuity of the specific cost function owing to the availability of only a finite number of cross sections. If there is a hinge but no support, point load, or cost discontinuity at B, then 3, V, and P are continuous across B and (3.19) reduces to (VO
+ vqB= 0,
(3.20)
where 8 and 8 are relative rotations associated with P and P, respectively. If there is a support but neither hinge nor cost discontinuity at B then (3.19) changes to (cBR + wR), = 0,
(3.21)
where and w are slopes and R and R reactions. If there is a discontinuity in the specific cost but no support, point load, or hinge at B, then (3.19) gives
3;
- I&
= 0.
(3.22)
Equations (3.20) and (3.22)were derived originally by Masur (1974, 1975). The optimization of elastic beams with various design constraints was considered also by Barnett (1961), Haug and Kirmser (1967), Prager and Taylor (1968), Sheu and Prager (1968), Prager (1968, 1970, 1971b), Shield and Prager (1970), Chern and Prager (1970),Chern (1971),and Dafalias and Dupuis (1972). General optimality conditions for elastic systems were proposed also by Masur (1970) and Mroz (1972). A number of papers have dealt with the optimization of beams with dynamic constraints but the latter class of problems is outside the scope of this article. IV. Optimal Flexure Fields-Basic
Geometrical Properties
A. DERIVATION OF OPTIMAL MOMENT-CURVATURE RATERELATIONS We have already established that (2.13) and (2.14) constitute necessary and sufficient conditions for the minimum weight of perfectly plastic beams of prescribed strength and elastic beams of prescribed strength or compliance if the cross sections are rectangular and their depth is constrained to
Optimal Load Transmission by Ffexure
207
a given value.? Considering an arbitrary point P of the middle surface of a grillage of such beams, we may start from a basic layout (Prager, 1974a.b) of potential beams in all directions. Then, (2.14) requires the rates of curvature to take on an absolute value not exceeding k in any direction. It follows that condition (2.13) for a nonzero beam moment may only be satisfied at P in directions of maximum or minimum curvature rates, which are termed principal directions (1, 2) and are known to be orthogonal. Therefore, the moment-curvature rate relations for point P can be characterized fully by xi = k sgn M i , for M i# 0, (4.11 1 K i 1 5 k, for M i= 0, (i = 1, 2) in which xi are kinematically admissible principal curvature rates having the same directions as the statically admissible principal moments M i . The relations in (4.1)are represented graphically in Fig. 8% in which the
(+> R+
o + o o s+
R-
s-
T
(b) FIG.8. (a) Optimal moment-curvature rate relation and (b) symbols for optimal regions.
principal moments M , and M , are the Cartesian coordinates, contour lines represent a constant value of the specific cost $, and vectors normal to the contour lines give the same rates of curvatures as (4.1). If MI 2 M , , then the t See Sections II,B, 111,B.and II1,C.
208
George I . N . Rozvany and Robin D . Hill
foregoing relation admits the following type of optimal regions in the so1ution:t
(A) R +
~1
(B) R-
(K1
=k,
1 Ik,
KZ=
= K, = k,
(AC) S+
ic1
(BD) S -
I C ~= K Z =
(AB) T
~1
=
-K,
(K11 I
(E) 0
I k,
12 .1
Ml 20,
-k,
MI 2 0,
M2=0,
MI =O,
M, 2 0,
-k,
M1 I0,
M , 5 0,
= k,
M1 2 0,
M, 5 0,
k,
Ik,
1 ~ 2 1
M , 50, (44
M,=M,=O,
where the letters in parentheses refer to points and line segments in Fig. 8a. If all points of a domain are subjected to a nonzero loading, then‘O-type regions of finite area cannot occur in the solution. In this article, R-type regions in figures are marked with arrows in one direction (Fig. 8b), S-type regions with small circles, and T-type regions with arrows in two directions. Arrows signify the directions of nonzero principal moments if they are determinate and the sign of principal moments is also indicated (Fig. 8b). Moment and curvature rate fields satisfying (4.1) are termed optimalJewre Jields. It is a point of interest that apart from the optimization problems considered, the same fields represent the solution to the problem of analysis of plates made out of ideal locking material (Prager, 1957) having the locking surface K~ = k ( i = 1, 2). In addition to the Prager-Shield condition, (4.1) may also be derived from theories of Masur (1970), Mroz (1972) and Save (1972), or by a direct variational method (Rozvany, 1974f).The same condition gives a minimum reinforcement volume for perfectly plastic fiber-reinforced plates of prescribed depth and strength (Morley, 1966). Since beam moments may only occur in principal directions and the specific cost function for all beams is t,b = k I M 1, where k is a known constant and M the beam moment, the optimal “cost” representing the minimum weight of the grillage becomes
*
Qmin=j(IM,I D
+
I M , I ) ~ ~ ~ Y ,
(4.3)
where M, and M, satisfy (4.1) and x and y are Cartesian coordinates. By (2.20) and (2.21) the minimum cost of the grillage may also be calculated t Symbols denoting the regions were proposed by Prager (1974a).
Optimal Load Transmission by Flexure
209
from either of the following expressions:
Qmin
=
I,
PU dx dy,
(4.5)
where M i and x i(i = 1, 2) satisfy (4.1), u(x, y) is the velocity field associated with the foregoing curvature rates, and P(x, y ) is the load. Obviously (4.3) gives an upper bound on the minimum cost for any statically admissible moment field and by (2.26), (4.5) gives a lower bound if the rates of curvature associated with u(x, y ) satisfy the condition
I K ~ I < k,
i = 1, 2.
(4.6)
B. PROPERTIES OF S- AND T-TYPEREGIONS In S-type regions both principal curvature rates take on the same value (k or -k) and therefore the rate of curvature is the same in all directions. Then, by (2.13), the directions of nonzero moments are indeterminate and only their sign is prescribed. If we replace K~ and I C with ~ - u ,and~ ~ in (4.2), then the curvature conditions for S-type regions furnish, after integration, the following general equations for the velocity fields u(x, y):
+ y 2 ) + a + bx + cy, u = &k(x2+ y2) + a + bx + cy,
S + : u = -&k(x2
(4-7)
S- :
(4.8)
where a, b, and c are constants. Next, it is shown that in R- and T-type regions, the centroidal axis of beams having a nonzero moment capacity is always straight. Shield (1960) has obtained the following relation for principal rates of curvature in plates: dKl/dS2 = (l/Pl)(K2
- Kl),
(4.9)
where s2 is a curvilinear coordinate measured in the direction of a line of principal curvature rate x2, and p1 is the in-plane curvature of the line of principal curvature rate K ~ . If M # 0, then by (4.1), icl = const (ie., k or - k). The derivative in (4.9) therefore takes on a zero value, and hence p1 = co for K~ # K ~ Q.E.D. .
2 10
George I. N. Rozvany and Robin D. Hill
This means that T-type regions consist of principal lines in constant directions at right angles, and the general equation for their velocity field is T: u = fk(y2 - x 2 ) + u T: u = kxy + a
+ bx + CY,
+ bx + cy,
(4.10) (4.11)
respectively, for principal directions parallel and at 45" to the coordinate axes x and y. The symbols a, b, and c denote constants. In T-type regions, the direction and signs of the principal moments are prescribed. However, for both T- and S-type regions, the foregoing conditions, in general, admit an infinite number of moment fields, which all yield the same optimal cost.
c. PROPERTIES OF R-TYPEREGIONS R-Type regions are less restricted kinematically than other types and therefore require a longer treatment. It is convenient to consider first a more general class of velocity fields that may be used for constructing R + regions. Defnition I . A velocity field u(x,y ) defined on D t E 2 is of Rt-type if (i) u(x, y ) is differentiable on the interior of the domain 6, and (ii) for each point P E D there exists at least one straight line 1 termed basic line with P E 1, such that along 1 n d and in the direction of 1 the curvature rate K of u(x, y ) takes on a constant value k. If at all points contained in 1 K = k is a principal curvature in the direction of 1, then 1 is termed a principal line. Defnition 2. For a given constant C, the contour lines K1 and K 2 contained in the domain D of an R+-type velocity field are defined by the relations y =f;(x) (i = 1, 2) such that u[x,f;.(x)] = C,
i = 1, 2.
(4.12)
Remarks. All R+-type regions contain an 8'-type velocity field, but the latter may form an R+-type region only if all its basic lines are principal Iines and the absolute value of the curvatures in a direction normal to basic lines nowhere exceeds k. In the propositions that follow, the terms basic line, principal line, and contour lines will always refer to lines in the domain of an R+-type velocity field. A trivial case of R+-type regions consists of parallel principal lines. If the latter are in the y direction, then the velocity field is given by u =f(x) + a + by - ky2/2 with 1 f x x I 5 k, where a and b are constants. R-type regions with nonparallel principal lines are termed fans.
Optimal Load Transmission by Flexure
211
It is known that at all points of a principal line the angular velocity (slope) in the direction normal to such a line is the same. Proposition 1 . A basic line 1 is a principal line if and only if it intersects K1 and K , at the same angle (a in Fig. 9a). Proposition 2. If 1 is a basic line and if at both PI E 1 and P , E 1 with PI # P , the angular velocity o in a direction normal to 1 takes on the same value, then 1 is a principal line. Proposition 3. The angular velocity co, along and in the direction normal is given by to a principal line W,
(4.13)
= k(FW),
't
1
(bf
~
N
P W
(Cf
(dl
FIG.9. Properties of R-type velocity fields.
George I . N . Rozvany and Robin D . Hill
212
in which the distance F W is furnished by the construction in Fig. 9a, where AW I K,, BW I K , , and AW = BW.? Proposition 4 . The curvature K, normal to a principal line AB (Fig. 9a) is given by
(4.14) with 4 = (f;'x+ 1 -lr,.xx) L o
Y
P =W
X
+ 1 +ffi.xx)
lx=o
9
where ( i = 1, 2 ) are the functions in (4.12), the coordinate Fig. 9a,f(O) =f1(0)= -f2(0), andXx(0) =f,,x(O) = -f,,x(O).
b =f(O),
(4.15)
is defined in
Proposition 5 . Considering a principal line D, the point N for which - k and the point Q for which IC, = - co can be found on the basis of the construction in Fig. 9c, in which p and q are furnished by (4.15).Another meaning of p and q is explained in Fig. 9d, in which AB and 71'8; are principal lines, A 4 -+ 0, and BPIAO = p/q.
K, =
Proposition 6. Let F and F' be points of maximum velocity along the and A", respectively, and let A 4 be the angle between AB and A B . Then for A 4 -0,
principal__ lines
z = o2A4/k,
(4.16)
where the distance z is defined in Fig. 9b and 0,is the angular velocity along Equation (4.16) implies that tan y = and in a direction normal to m2/kr(4) in Fig. 9b.
a.
Dejnition 3. The point W (Fig. 9a) with AW = BW, A E K , , B E K , , AW I K , , BW I K , , on K , and K, u = const, is called an intercept point of the principal line 3. The locus of all intercept points of an R'-type field is termed the intercept line. By Proposition 3, the intercept line is the same for any pair of contour lines of an R'-type field. Proposition 7. FW having the properties FW I tangent to the intercept line at W (Fig. 9a).
a, max,,
u = u ( F ) is a
Dejnition 4 . The curve that is contained in the domain of an R'-type field and to which all principal lines are tangent is termed the envelope of the 17'-type field. Point Q in Fig. 9c is contained in the envelope. t The symbol I indicates that two lines or curves are orthogonal at their intersection.
Optimal Load Transmission by Flexure
213
Instead of expressing velocity fields u( ) in R-type regions in Cartesian coordinates (x,y), it is often convenient to adopt coordinates (4, <) similar to those used by Shield (1960), in which 4 signifies the direction of a principal line ( A B in Fig. 9b) and represents the distance of an arbitrary point P from the point (F in Fig. 9b) of maximum velocity along the principal line. The distance of the latter point (F) from the envelope (LM in Fig. 10a) is
<
(a)
FIG.10. Representations of 8-type velocity fields.
denoted r ( 4 ) (see Figs. 9b and 10a).If the maximum velocity along a principal line is a(+), then the velocity at other points is given by
u = a(&) - kr2/2.
(4.17)
It follows from the foregoing discussion that two different representations are possible for an R+-type field. 1. First Representation An R+-type velocity field is determined fully if the following are specified:
(i) the envelope (LM in Fig. lOa), i.e., the curve to which all principal lines are tangent; (ii) a function 02(4)defining the variation of the angular velocity in a direction normal to the principal lines; (iii) along one arbitrary principal line (MN in Fig. 10a) defined by 4 = 4,, the maximum velocity a = ii, (at F, in Fig. 10a) and its distance ro from LM measured along MN.
<
Figure 10a also indicates that the curve F , F F representing = 0 and the curve F,H having the same directions as the principal curvatures x2 and
214
George I . N . Romany and Robin D. Hill
containing Fo are not the same. Their distance ~ ( 4measured ) along a principal line and the maximum velocity along the latter are, respectively, furnished by (4.18)
2. Second Representation u( ) in R+ may also be determined fully by specifying the curves K 1 and
X, (Fig. 9a) of constant velocity (C) and using the construction in Fig. 9a with AW = BW for finding the principal lines AB. The velocity along AB is given by
u = C - k[12 - (AF),]/2.
(4.20)
The properties of R+-type velocity fields presented herein apply equally to R--type fields except that in all statements containing the symbol k the latter must be replaced with - k. In constructing R-type regions, it is necessary to exclude those subsets of the domain of R-type velocity fields for which the absolute value of the curvature rate in a direction normal to principal lines exceeds k. D. PROOFOF PROPOSITIONS 1-6
1. Proof of Proposition 1 Suppose that fl(x) and f2(x) in (4.12) are represented by the following Taylor series expansions about x = 0:
fl(x) = b + ax f2(x) = - b
+ cx2 + O(x3), - (a + + ex2 + 0(x3), E)X
(4.21) (4.22)
where O(x3) denotes terms of third or higher degree. Points B, A, B’, and A’ in Fig. 10b have the following coordinates: B(0, - b), A(0, b),
B’[Ax, -b - (U A‘[A Ax, b
+ E ) AX + O(Ax2)],
+ aA Ax + O(Ax2)J.
(4.23)
The angular velocity at E in the direction of the x axis is
IE
u , ~ = lim[u(E‘) - @)]/As,
(4.24)
Optimal Loud Transmission by Flexure
215
in which u(E) = &(b2 - t;’)
+ C,
(4.25)
As = ( Y E - Y B c ) ( x A t - x f f ) / ( y Ar YB!) f
xBc
-xB
(4.26)
9
+
+
u(E’) = $k{$(A’B’)’ - [t; - f(yAt Y , . ) ] ~ ( A ’ B ’ ) ~ / ( ~yA, ), ’ } C , (4.27) where the factor (A’B’)’/(y,, - y,,)’ aliows for the fact that A’B’ is not parallel to AB, As is normal to AB, and f ( y A ,+ y S ) is the y value of F’. Using (4.23), (4.26) becomes AS = AX(((^ - l)/b
+ A + 1}/2 + 0 ( A x 2 ) .
(4.28)
Now
since ( x A ,- xB,)’ = O(Ax2).Further, ( A ’ S ) , = ( y A ,- yS)’
+ O ( A x z )= 4b2 + 4b Ax(ul + a + + O(Axz). E)
(4.30) Hence (4.27) yields
u(E‘) = fk{bz - t;’
+ A x [ < a ( l- I ) + &(A + 1) + E(b - <)I} + C + O(Axz). (4.31)
Then
u(E‘) - u ( E ) = fk Ax[c(b - t;) + t;a(A - 1) + a b ( l + l ) ] + O(Ax2). (4.32) Using (4.24), (4.28), and (4.32),
IL
E(b - t;)
‘J= ku6/u[t;(l- 1) + b ( l + l)]
+
l)‘
(4.33)
When E = 0, u , is~independent of 5 and therefore the line AB is a principal direction. Further, it is easily seen from (4.33) that u , is~ independent of t; only when E = 0. Thus it has been shown that the condition E = 0, which is equivalent to the requirement that curves K , and K, intersect the line AB at equal angles, is necessary and sufficient for the principality of AB. 2. Proof of Proposition 2 Further information can be obtained from (4.33) by noting that there is a ~ respect to when e, I, and b are strictly monotonic variation of u , with nonzero. If the slope normal to a basic line is the same at any two distinct
<
216
George I . N . Romany and Robin D . Hill
points, then E must be zero if 1 and b are nonzero and by Proposition 1,E = 0 implies principality.? 3. Proof of Proposition 3
For E = 0, (4.33) gives (4.34)
u,x= kab,
which will be denoted o2and is numerically equal to k(FW), where F W is indicated in Fig. 9a. This is because by (4.21), b = AF and a = tan /3. 4. Proof of Proposition 4
At E’ in Fig. lob, the angular velocity in the x direction is denoted uVxIE, and the angular velocity in the direction of A’H and normal to A’B’, by o1I E , and o2I E , respectively. Then the following relation holds : u,x
IE,
IF,
IE,)
cos Aqi - (a1l E , ) sin A$. = k. Hence
(4.35)
w1lE = k(E’F‘).
(4.36)
= (a2
Clearly, o1 = 0 and rcl
Then o2lE, can be calculated by applying (4.34) to A‘B: o2
Further,
lE,
= kab
+ k[ja2(1+ 1) - (e - k ) b ] A x + O(Ax2).
sin Aqi = Aqi + O(Aqi2) = (Ax/2b)(l- 1)
+ 0(Ax2),
(4.37) (4.38)
t If I = 0, in which case the basic lines have identical velocities at their intersection, then (4.33) implies principality because u , becomes ~ independent of l.In general, the velocity u is not the same for two principal lines at their intersection if A 4 is finite, because the principal lines give valid velocities for a given R’-type field only on one side of their intersection with the envelope. In Fig. 10% for example, the principal line F L gives the correct velocities for the R+-type field considered only up to the point L, and its extension beyond that point (with a curvature rate k ) gives a different velocity u at the intersection of F L a n d N M from the value of u defined along N M . However, when A# +O, then the distance between the envelope and the intersection of the two principal lines enclosing A# is O(A#’), and hence for finite angular velocities of u( ) the difference between the u values for the two principal lines at their intersection is also O(A4’). This means that the variation of u along such principal lines (e.g., AB and A’S in Fig. 9b) can be expressed as u = a + br - kr2/2 (on AB), u = a + cr - kr2/2 + O ( A @ ) (on A’S),where a is a constant and b and c are the angular velocities in the directions of AB and AS at their intersection. It follows that the angular velocity in the direction normal to AB at an arbitrary point F of AB is { ( c - b)r + O(A4’)}/r A 4 , which gives (c - b ) / A 4 = const after neglecting infinitesimals of higher order. This is because ( c - b ) itself is O ( A 4 ) if the velocity field is differentiable on Int D. It follows that a / / basic lines of an 8’ region are principal lines if and only if the first derivatives of u( ) are bounded on the envelope (Bd b). The extension of an 8’ region beyond its envelope is discussed on page 278.
Optimal Load Transmission by Flexure
217
and E'F' = 5 - i(yAt+ y p ) + O(AxZ), =5
- $[u Ax(l - l)] + 0(Ax2).
(4.39)
Combining (4.35)-(4.39), we get
IE,
u , ~ = kab
+ k Ax[a2(l + 1)/2 - b(e - l c ) + (1 - l)5/2b] + O(Ax2) (4.40)
Then u2 can be calculated from (4.24) and (4.40):
lE, - u , IE)/As ~ - l(a2 + 2bc -
-u2 = u,xx= lim(u,,
A(t/b
+ 1) + 1 -
(4.41)
For principal lines, however, 1 takes on a specific value owing to the condition 61 = 6 2 in Fig. lob, in which
+ 2cl Ax) + A$,
(4.42)
= tan- '(a - 2e Ax) - A+.
(4.43)
tI1 = tan- '(a 62
By replacing sin A$ with tan Aq5 in (4.38), the condition 8' = 8, yields tan- '(a - 2e Ax) - tan- '(a
+ 2cl Ax) - Ax(l - l)/b = 0,
(4.44)
which gives
l = (a2
+ 1 - 2be)/(a2 + 1 + 2bc) = 4/p.
(4.45)
Substituting (4.45) into (4.41), after simplification we get (4.14) and (4.15), in which
f (0) = b, fl,xx(o) = 2c3
XX(0) = a, f2,XX(O)
= 2.
(4.46) (4.47)
5. Proof of Proposition 5 Substituting
I C ~=
-k into (4.14) yields
5 = b(P4 - 4 - PM4 - P).
(4.48)
By considering similar triangles in Fig. 9c, we get b ---
NF
4(P - 4)- f(P + 4 ) - 4P4 ' which gives the same value for NF = 5 as (4.48).
(4.49)
George I . N . Rozvany and Robin D. Hill
218
6. Proof of Propositions 6 and 7
The y value of F is yF, = a Ax(A - 1)/2 + O(Ax2).
+ O(Ax) = ab + O(Ax), so that = ab sin A+ + O(Ax2) = a Ax(A - 1)/2 + O(Ax2).
(4.50)
Now F'W' = FW y F ,- y,,
(4.51)
The effects of (4.50) and (4.51) therefore cancel each other out and thus there is no first-order difference between the y values of W and W'. This means that FW is the tangent to the intercept curve at W. The distance z in Figs. 9b and 11 can be expressed as z = YF,
()(AX2)
(4.52)
Equations (4.34) and (4.38) imply
IAB
o2
A+/k = a Ax(A
-
1)/2 + O(Ax2).
(4.53)
Then (4.50), (4.52), and (4.53) give (4.13) after neglecting infinitesimals of higher order.
FIG.11. Proof of Proposition 7.
Proposition 6 was derived elsewhere by a more direct method (Rozvany, 1974e) and most other propositions presented herein were obtained in an earlier paper (Rozvany et al., 1973). Proposition 2 was also stated by Lowe and Melchers (1974b), who used a different proof.
E. TOPOGRAPHICAL PROPERTIES OF OPTIMAL FLEXURE FIELDS In Sections II,B and C we discussed the geometrical properties of various types of optimal regions without taking the boundary conditions (constraints) into consideration. Additional geometrical properties of optimal
Optimal Load Transmission by Flexure
2 19
flexure fields can be derived, making an allowance for kinematic constraints on the velocity field u( ). Along lines of simple supports (S,) and clamped supports (S2), for example, the following constraints must be observed : on S,:
u =0,
on Sz:
u = 0,
u ,=~0,
u , =~ 0,
(4.54)
where x and y are Cartesian coordinates along the middle plane of the grillage. Subsequent discussion herein and in Sections V and VI will be restricted to the following class of problems: (i) The load is nonnegative, P(x, y) 2 0, i.e., downward on the domain D. (ii) All boundaries of the domain D are either simply supported or clamped. In order to examine various characteristics of optimal flexure fields in an example, a relatively complex solution is shown in Fig. 12, in which the system has a clamped support along the external boundary (thick lines) and at the point P. The domain in Fig. 12 is divided into two subsets, which shall be termed branches (unshaded areas) and junctions (shaded area). The so-called center-
FIG.12. Example of an optimal flexure field.
220
George I . N . Rozvany and Robin D . Hill
lines of branches are shown in dash-dot lines. Considering the velocity field u( ) along one principal line only, the former takes on its maximum value at the centerline. Each branch is divided by the centerline into two subsets, which may fall into one of four categories termed a, fl, y, and 6 j e l d s . According to the two fields they contain, branches are of ten different types termed aa, as, ay, ad, flfl, fly, P4, yy, yh, and 66 type. fl Fields are always associated with clamped boundaries and a, y, and 6 fields with simply supported boundaries. The difference between optimal regions (R', S', etc.) and optimal fields (a, fl, etc.) is that the former are based on curvature conditions [(4.1) and (4.2)] only and the latter take boundary and kinematic continuity conditions also into consideration. Only clamped supports (flp branches) are considered in Section V and mixed support conditions are discussed in Section VI. The foregoing topographical properties were first discussed by Rozvany (1973f). Before outlining the general theory for the foregoing problems, axially symmetric optimal flexure fields will be considered briefly.
F. AXIALLYSYMMETRIC SUPPORTS AND LOADING Optimal flexure fields for clamped and simply supported circular domains-in the particular context of fiber-reinforced plates-were first obtained by Wood (1961) on the basis of the uniform energy dissipation principle (Drucker and Shield, 1956), and solutions for other axially symmetric boundary conditions were derived later by a purely static method (Rozvany, 1968; Rozvany and Melchers, 1970). Other authors considered the effect of the variation of the moment arm (Mroz and Shamiev, 1970) and nonconstant plate thickness (Melchers, 1973) on the optimal moment field. In this section, solutions are derived from the optimal moment curvature relations (4.1) in which the subscripts (1, 2) are replaced by (0, r), denoting polar coordinates. Since the principal curvatures become
- u,rlr,
(4.55)
,
(4.56)
KO
=
Kr
= -u,rr
it follows that for a finite annular segment (II < r < r 2 ) K~ = k = const
because (4.55) with
K~ = k
implies
K, = k,
(4.57)
gives u = -kr2/2+ C,
(4.58)
Optimal Load Transmission by Flexure
I
22 1
Ilk\
I
FIG. 13(a)-(d).
Optimal velocity fields for axially symmetric supports and loading.
222
George I . N . Romany and Robin D . Hill
FIG.13(e)-(l)
Optimal Load Transmission by Flexure
(r)
FIG. 13(mt(r)
223
SUPPORT CONDITIONS
OPTIMAL REGIONS
SUPPORT CONDITIONS
@g
-L:I2L-L/4
01
OPTIMAL REGIONS
@ '._. R-
Optimal Load Transmission by Flexure
225
and then (4.56) and (4.58) yield K, = k. This means that only the following regions may occur in the solution. (i) S-type regions S’:
ice = IC, =
k,
Me 2 0,
S- : K O = K, = - k, (ii) R-type regions R’:
K,=
lice(
k,
R-: ~ , = - k ,
M , 2 0,
Me S O ,
S k,
( ~ g / < k ,
M,>O, M,SO,
M, S O .
M,=O, Mo=O.
(4.59) (4.60) (4.61) (4.62)
As was indicated earlier, the optimal moment field is usually nonunique in S-type regions. Optimal velocity fields satisfying (4.1) for nonnegative (downward) loads are shown in Fig. 13 and the corresponding solutions are given in Fig. 14. Here, and throughout this article, thick lines in plan view denote clamped edges, double lines simple supports, and single lines free edges. Solutions in Fig. 13% b, e, f, h, i, j, k, 1, m, and n are similar to the corresponding solutions for optimal beams obeying Heyman’s (1959) condition (2.13) and therefore require no further explanation. The value of p in Fig. 13c may be calculated by considering the extension (broken line) of u(r) to the axis of symmetry,
~ ( 0=) - kpi/2,
(4.63)
and thus the curvature rates indicated give the following condition for u(p) = 0 :
-Pi/2
+ Pl(P - P
d ) - (P - P1I2/2 = 0,
(4.64)
or p
- p1 = (p2/2 - p6/2)”2.
(4.65)
When po = p/3, (4.65) gives p1 = p/3 and thus no R, region occurs in the solution. Hence for po p/3, the solution becomes the one shown in Figs. 13d and 1 4 . The validity of the latter solution may be checked by the construction shown in Fig. 13d, in which the point P must lie inside the inner support Q for the solution indicated. It can also be checked on the basis of (4.14) or the construction in Proposition 5 (Fig. 9c) that u2 = -k along the inner support if po = p/3. FIG. 14. Optimal solutions for axially symmetric supports and loading. In S + regions, in S- regions K, = = - k ; in R C regions K, = k and in R - regions K, = - k .
K, = K~ = k ;
226
George I . N . Rozvany and Robin D . Hill
The position of the optimal region boundary in Fig. 14g may be derived from the boundary conditions ~ ( 0= ) -p:/2,
u,,(O) = 0,
~ ( p=) 0,
u,,(p) = 0.
(4.66)
The region boundaries in Fig. 14a-n do not depend on the distribution of a nonnegative load P ( r ) 2 0. However, due to simple supports on the interior of the domain D, some region boundaries are dependent on P ( r ) in Fig. 140-r. It can be seen in ~ the simple support is kinematFig. 130, for example, that the slope u , at ically indeterminate, but it becomes unique if equilibrium together with conditions (4.1) are also taken into consideration. For further details and applications of the solutions listed herein, the reader is referred to earlier papers (Rozvany, 1968; Rozvany and Melchers, 1970). It was pointed out correctly by Melchers (1975a) that region boundaries for axisymmetric domains without internal supports (Fig. 14a-n) are valid for nonaxisymmetric loads also. In fact, all optimal velocity fields discussed in Sections V and VI are known to be (Rozvany, 1972,1973g)independent of the choice of a nonnegative load P( ) 2 0. V. Optimal Flexure Fields-Clamped
Boundaries
A. REVIEWOF LITERATURE ON NONAXISYMMETRIC OPTIMAL FLEXURE FIELDS
Early theories for fiber-reinforced plates of minimum reinforcement content were proposed by Mroz (1958) and Kaliszky (1965). The optimal moment-curvature rate relations in (4.1) were obtained independently by Morley (1966) and Mroz (1967), in the context of reinforced concrete slabs. The optimal flexure field for square simply supported domains was derived by Morley (1966). The same solution was presented earlier for slightly different design constraints by Rozvany (1966a,b). Morley (1966) also obtained the correct solution for simply supported rectangular domains. Reitman (1967) and Clyde (1967) confirmed by linear programming the analytical optimal solution (Rozvany, 1966a) for square simply supported domains. Sacchi and Save (1969) derived an optimal solution for elliptic simply supported plates. From static (upper bound) considerations and without a proof of global optimality, Melchers (Lowe and Melchers, 1972) obtained the correct solutions for square and polygonal clamped boundaries and for regular simply supported triangular boundaries. The validity of Melchers’ solutions was soon proved on the basis of purely kinematic considerations arising from (4.1) (Rozvany and Adidam, 1972b).
Optimal Load Transmission by Flexure
227
The optimal moment-curvature rate relations in (4.1) yielded solutions and methods for rectangular boundaries with various support conditions (Romany and Adidam, 1972c), clamped boundaries of arbitrary shape (Romany, 1972, 1973e,f), simple supports (Romany e! al., 1973), mixed boundary conditions (Romany, 19738, 1975b), corners (Romany, 1974g), interior supports and alternate loading conditions (Romany, 1974d, 1973g), and edge beam supported domains (Lowe and Melchers, 1974a).
B. PROPERTIES OF SOLUTIONS FOR CLAMPED BOUNDARIES Topographical properties of optimal flexure fields were discussed in Section IV,E. The particular case of clamped boundaries (bp branches) will be considered first because of the comparative simplicity of their geometry. pfl Branches are bounded by two clamped edges ( d , and d , in Fig. 15) and have the following properties.
FIG. 15. Property 1.
Property 1 . A BB branch consists of one inner and two outer regions. In general, the outer regions are of R- type and the inner region of R+ type (which may degenerate into S - and T - or S+-type regions, respectively; see Properties 4 and 5). The region boundaries (b, and b, in Fig. 15) between the inner and outer regions and the centerline h may be determined on the basis of the construction in Fig. 15 in which C A = AW = EB = BW;
228
George I . N . Rozvany and Robin D. Hill
AF = BF; CW I d , ; EW I d , ; A E b , ;B E b,; F E h ; K , = - k on C A and BE; and K~ = k on AB. W is termed the intercept point. The construction in Fig. 15 is subject to the restrictions CW < r ( d l ) c , EW < r(d&,
for r(dl)c > 0, for ~ ( d , > ) ~0,
(54
where r( ) denotes the radius of curvature of the domain boundary and the sign convention for r( ) is given in Fig. 16. d
d
6
(b)
(a)
FIG. 16. Sign convention for radius of curvature of domain boundaries.
Property 2. A branch centerline ( h in Fig. 17) and the region boundary (b in Fig. 17) may only have their intersection G at a distance r(&/2 from the point H of locally maximum boundary curvature, measured at right angles to the domain boundary d. d
FIG. 17. Property 2.
Property 3. If the intercept point (W in Figs. 15 and 18) is at an equal distance from n boundary points with n > 2, then the inner region changes into an n-sided S+-type region in the vicinity of such intercept point, and its boundaries can be constructed on the basis of Fig. 18 (which shows the specific case n = 3 ) .
Optimal Load Transmission b y Flexure
229
FIG. 18. Property 3.
The foregoing S+-type region is termed a junction of n /?/?-typebranches and the point W is called the center of the junction. If W is contained in the junction, then it represents a local maximum of the velocity field u( ). Remark. Property 2 can be used for locating the outer end (points A, B, C, D, and E in Fig. 12) of centerlines of branches, which can be used as a starting point for the construction of branches on the basis of Property 1. The inner end of branches (points I, J, K , L in Fig. 12) is found by employing Property 3 and further branches may initiate from such inner ends (branches ZJ, JK, and K L in Fig. 12). Branches with at least one outer end (on the domain boundary) are termed exterior brunches (AZ, BZ, CJ, DL, and EL in Fig. 12) and branches with two inner ends interior brunches. A pair of junction centers (e.g., K and L in Fig. 12) may be connected by two interior branches. The boundary curvature (l/r) takes on a value - co at F and G and CQ at A, B, and D. A local maximum of the curvature with r = r/2 = 0 occurs only at the latter three, and hence by Property 2, the region boundary is at a zero distance from the domain boundary at A, B, and D.At C and E in Fig. 12, r/2 takes on a finite value.
+
Property 4 . The outer regions degenerate into S--type regions around reentrant corners and clamped-point supports, and the inner region degenerates into a T-type region between two such S--type regions. The foregoing regions can be constructed on the basis of Fig. 19. Boundaries between S - and R--type regions are normal to the domain boundary segments adjacent to the reentrant corners.
230
George I . N . Romany and Robin D . Hill
FIG.
19. Property 4.
Property 5 . When an intercept point ( W in Fig. 20) is at a constant distance r from a circular domain boundary segment, then the inner region degenerates into an S+-type region (Fig. 20) and the centerline intersects the circular domain boundary segment at its midpoint (E in Fig. 12).
\ FIG.20. Property 5.
Property 6. If a fifi branch is associated with a clamped point or reentrant corner and a straight boundary segment at a distance a (Fig. 21) from each other, then the region boundaries are parabolas:
and the centerline (dash-dot line), intercept line (broken line), and envelope
Optimal Load Transmission by Flexure
23 1
-x
FIG.21. Property 6.
(dash-double dot line), respectively, are given by y , = a/2 + 4x:/9a,
yw = a/2 + x&/2a,
7 = 1.25a - 3(Js/4)2f3a1f3,
(5.4) where x and y are the Cartesian coordinates shown in Fig. 21. The principal lines in the inner (R') regions, are normal to the parabola given by (5.3). [Properties 1-6 were obtained by Rozvany (1972).]
c. PROOF OF PROPERTIES 1-6 1. Proposition 8 Let the velocity field u( ) be defined on D such that u( )is continuous on D and is differentiable on the interiors of R, c D, R, c D, and R , c D. Further, let the points A and B be contained in the common boundaries of R , / R 2 and R , f R 3 ,respectively, and let (Fig. 22a)
(i) the curvature rate on AB c R, be K = k; (ii) the angular velocities at A and B in R , and R J , respectively, be
o , / k = AW,
o , / k = BW,
aA= 0, O B = 0, where F E AB with FW I a (Fig. 22a);
(5.5)
(5.6)
U A - u s = (w: - Wi)/2. (5.7) (iii) Then the slope is continuous across the region boundaries at A and B.
232
George I . N . Rozvany and Robin D . Hill
(b) FIG.22. Propositions 8 and 9.
2. Proof of Proposition 8 Considering the velocities at A, M, and N with ( A M )-P 0 in Fig. 22,
+ o A ( A M ) + O(AM)’, UN = u y + O(MN)’,
UM = U A
(5.8) (5.9)
and hence u,? I A
cos 4,
= (UN - u&AN = wAAM/AN =
on R , .
(5.10)
Similarly, u , IB~ = oBcos rl,
on R3 ,
(5.11)
Next, it is shown that max u = u(F).
(5.12)
AB
If (5.12) holds, then by condition (i) and Fig. 22b, UA
- U S = +k[(AW)’ COS’ 4 - (BW)’ COS’ q].
(5.13)
However, by the sine rule (AW)sin (b = ( B W )sin q,
(5.14)
Optimal Load Transmission by Flexure
233
which implies (BW), cos2 q = (BW)Z - (AW)2(1 - cos2 4).
(5.15)
Substituting (5.15) into (5.13) and making use of (5.5), we get
(5.16)
Since the value of (uA- us) uniquely determines the location of u,,, along AB with K = k, the identity of (5.7) and (5.16) proves (5.12). Then it follows from Fig. 22b and (5.5) that u , IA~ = k(AW) cos
4 = O ~ ' C O S 4,
on R , ,
(5.17)
u , lg~ = k(BW) cos
= wg cos q,
on R2 .
(5.18)
Equations (5.10), (5.11), (5.17), and (5.18) establish slope continuity at A and B in direction AB. At A and B, the slope is also continuous in the direction of the region boundaries and then, due to slope continuity on the interior of R1, R, ,and R , , the slope is also continuous in any arbitrary direction.
3. Proposition 9 If AB in Fig. 22 is a basic line (see Definition 1) then it is a principal line. 4. Proof of Proposition 9
In the direction normal to AB, the angular velocities at A and B, respectively, are 0: = w A sin
4,
(5.19)
wz = w g sin q.
Then by (5.5) and Fig. 22, (5.20)
w: = k(FW) = o;,
and thus Proposition 2 implies that AB is a principal line.
5. Proof of Property 1 The construction in Fig. 15 satisfies all requirements of Proposition 8 with = k(AW) = m g = k(BW),
UA
-
= 0.
(5.21)
The restriction a,= a,= 0 follows from the fact that at C and E the angular velocity w is zero in all directions, and CA and EB are principal lines along which the angular velocity (slope) is constant in the normal direction. The principality of CA and EB can be shown by replacing each clamped boundary (e.g., dl in Fig. 15) with two simply supported lines (d;and d'; in
234
George I . N . Rozvany and Robin D. Hill
FIG.23. Proof of Property 1.
Fig. 23) at a constant distance 6 and then applying Proposition 1 to the simple supports. The condition of principality in Proposition 1 does not depend on 6 and thus it still holds when 6 -+ 0, giving the same kinematic constraint as a clamped edge. The limitation (5.1) follows I'rom (4.14) and (4.15) and Fig. 23, in which points C' and C" correspond to A and B in Fig. 9a. The quantities in (4.15) take on the following values:
Then (4.14) gives for S
0 (5.23)
which gives K , < - k for 5 > r(dl)c/2 and hence at A (4.1) is violated when the requirements in (5.1) are not met. A different proof of (5.1) was given elsewhere (Rozvany, 1973f). It follows from Proposition 9 that AB in Fig. 15 is also a principal line. This concludes the proof that the velocity field u( ) generated by the construction in Fig. 15 ensures kinematic admissibility and satisfies (4.1) if the corresponding moment field (Ml, M,) is statically admissible. The latter condition can always be satisfied for a nonnegative (downward) load P( ) 2 0, as can be seen from Fig. 24, in which s is the beam spacing and F A
Optimal Load Transmission by Flexure
235
(b)
(a)
FIG.24. Beam moments in (a) outer and (b) inner regions.
and F , the forces transmitted from the “suspended” beam AB to the “cantilever” beams C A and EB. 6. Proof of Property 2
Considering the boundary configuration in Fig. 25a, it follows from (5.1) that a clamped circular domain boundary segment cannot be adjoined by another circular segment of greater curvature. If a -+ 0, fl --t 0, in Fig. 25a
(a)
FIG.25. Proof of Property 2.
then the same diagram indicates that the curvature cannot increase as we move away from the optimal centerline. Hence the centerline must intersect the domain boundary at a point of locally maximum curvature. However, the converse of the foregoing statement is not always true, that is, not all boundary points of maximum curvature represent an outer end of a centerline. In Fig. 25b, for example, the boundary segment CT is adjoined by boundary segments of smaller curvature, and yet no centerline intersects the boundary segment because the distance C W is smaller than the radius of curvature r. The same would apply if CT had a nonuniform curvature but the radius of curvature at the local maximum curvature exceeded the corresponding distance CW. The distance of points H and G in Fig. 17 can be determined by applying the construction in Fig. 15 to domain boundary points on both sides of H
236
George I. N. Rozvany and Robin D. Hill
H
r(d),+/2 . I
FIG.26. Proof of Property 2.
and then taking the limiting case for A 4 -+ 0 (see Fig. 26), giving
+ H F = H G + GF = $WC + O(A@)
WC = ~ ( d ) O(A@), ~
=fr(d)H
+ O(A@).
7. Proof of Property 3
Using the construction in Fig. 18 for determining the region boundaries of a junction of fifi branches, it follows from Proposition 3 that in the R+-type regions around the junction the angular velocities along and in a direction normal to the sides AB, AZ, and BZ in Fig. 27 take on the values k(FW),
% FIG.27. Proof of Property 3.
Optimal Load Transmission by Flexure
237
k(F‘W), and k(F”W), respectively. If the junction consists of an S+-type region with the maximum velocity at Wand we adopt Cartesian coordinates as shown, then (4.7) gives 0
= -u,xx
IxqW
= k(FW).
(5.24)
This establishes slope continuity along AB and similar conclusions may be obtained for A Z and BZ by an appropriate rotation of the coordinate axes.
8. Proof of Property 4 Considering the two reentrant corners in Fig. 28, along the line AB the slope in the normal direction is (Proposition 3) u,x I A 8 = k ( W .
(5.25)
FIG.28. Proof of Property 4.
Since xF = (TF) + r sin A+ = (TF)+ O ( r ) = ( F W ) + O(r), for r 40, u,x JAB
= kx \ A 8
7
(5.26)
and thus K~ = - u , , ~= - k . The region boundaries follow from the construction in Fig. 15.
9. Proof of Property 5 For a clamped circular boundary segment, the construction in Fig. 15 gives a constant velocity u = C along the region boundary b (Fig. 29) and hence the latter is a contour line. Then the quantities in (4.15) become (Fig. 29) J,(O) = cot a, f(0) = ( r sin a)/2, (5.27) f,,,,= -f2,xx = -2(1 + cot2 ~ t ) ~ ’ /= ” / -2/r r sin3 a,
238
George I. N.Rozvany and Robin D.Hill
FIG.29. Proof of Property 5.
and hence p = q = 1 + cot’ a - l/sin’ a = 0.
(5.28)
Then by (4.14) ~2
= k,
(5.29)
giving an S+-type region along AB. 10. Proposition 10 Let an S-type region and an R+-type region have a common boundary b. Then the principal lines in the foregoing R+-type region are normal to b. 11. Proof of Proposition 10
Considering a point P E b, the curvature in a direction that is tangent to b takes on a value - k. Then by (4.1), any tangent to b is a principal direction for the velocity field in the adjacent R+-type region. Due to the orthogonality of the two principal directions, the principal line containing P is normal to b. 12. Proof of Property 6 In Fig. 21, by the Pythagorean relation,
- a)’ ( 2 ~=~(2yB ) ~
+ xi ,
(5.30)
Optimal Load Transmission by Flexure
239
which gives (5.2). Further, EW = 2yA- a,
(WC)’ = 2’(y, - a)’
+ 2’x2,,
(5.31)
and hence the condition E W = C W reduces to (5.3). Equation (5.4) follows readily from (5.2), (5.3), and the relations (see Fig. 21) y~ = ( y ,
+ yB)/2,
XF
= 1.sxA = 0.75xB.
(5.32)
The second equation (5.4) can be derived from the relations (Fig. 21) yW=2yB,
xW=xB.
(5.33)
The orthogonality of the principal lines in the inner region (R’) to the region boundary given by (5.3) follows from Proposition 10. The equation of the envelope can be obtained by adding to the x and y values in (5.2) the coordinates of the radius of curvature for the curve represented by (5.2). D. EXAMPLES 1. Convex Polygonal Domains Using the foregoing properties, solutions may be derived readily for triangular (Fig. 30a), quadrilateral (Fig. 30b-e), and other polygonal (Fig. 30f-g) domains. It can be seen from Fig. 30b-e that quadrilateral domains may contain one or two S+-type regions (junctions). The optimal solution for regular n-sided polygonal domains (Fig. 30f) contains n R--type regions, n R+-type regions, and one S+-type region. In the case of irregular polygonal boundaries, the S+-type region splits into several parts separated by R+-type regions (Fig. 30g). 2. Domains Consisting of the Complement of Bounded Sets In Fig. 31, solutions are given for point supports (Fig. 31a-b), rectangular supports (Fig. 31c), a combination of octogonal and square supports (Fig. 31d), cross-shaped supports (Fig. 31e), circular supports (Fig. 31f,g), and square supports having diagonals parallel to the grid lines (Fig. 31h). In Fig. 314 the region boundaries in polar (8, p ) and Cartesian (x, y ) coordinates are given by
+ b], x = 3(a tan 8 + b sin 8),
(5.34)
p = $[(a/cos 0)
y = f(a + b cos 0).
(5.35)
240
George I . N . Rozvany and Robin D. Hill
(el (Cl
FIG.30. Solutions for clamped convex polygonal domains.
c
I
b
(a)
X
I,
-3b
,
*I
(g) FIG.31. Solutions for the complement of bounded sets.
242
George I . N . Rozvany and Robin D. Hill 3. Combination of Straight-Line and Point or Circular Supports
Figure 32 gives solutions for a rectangular domain with a point support (Fig. 32a), combination of point and line supports (Fig. 32b), and circular and line support (Fig. 32c). Curved lines in Fig. 32a,b are given by (5.2) and (5.3). The value of b in Fig. 32b can also be calculated on the basis of (5.2), in which x B = a12 giving b = 5a/16. (5.36) .12
,
./2
,
.
I
,
./2
, r
.I2
,
FIG. 32. Combination of line and point or circular supports.
In Fig. 32c, the region boundaries may be derived in polar coordinates (0, p ) from the relation z = (a - R cos 0)/(1
+ cos 0).
(5.37)
4. Polygonal Domains with Reentrant Corners
Figure 33 shows the solution for L-shaped (Fig. 33a), T-shaped (Fig. 33b), and cross-shaped domains (Fig. 33c).
Optimal Load Transmission by Flexure
,"
243
(b) (a)
(C)
FIG.33. Solutions for domains with reentrant corners.
In Fig. 33a,
b = (a - c)/2.
(5.38)
Substituting xe = c, y e = b, (5.2) and (5.38) yield c = ($
- l)a,
b = (2 - $)a/2.
(5.39)
The value of b in Fig. 33b is furnished by (5.37). 5 . Domains Consisting of the Union of a Half-Space and Bounded Set
Figure 34 shows solutions for a combination of half-space and semicircle (Fig. 34a), triangle (Fig. 34b), rectangles (Fig. 34c-d), rectangle and two quarter-circles (Fig. 24e), half-ellipse (Fig. 34f), triangle and trapezium (Fig. 34g), and a solution obtained by superposition (Fig. 34h).
244
George I . N . Romany and Robin D. Hill
(h)
FIG.34. Solutions for the union of a half-space and a bounded set.
6. Domains Having Curved Boundaries
For a symmetric curved boundary y = k f ( x )(Fig. 35a), the region boundaries on the basis of Property 1, Fig. 15, are given by
7 = y/2,
R =x
+ y,,y/2,
Y,x = Y,,/(2 + Y Y , x x + Y
3.
For a parabolic boundary (Fig. 35b) with y = _+xl/’ (5.40) yields j j = (x - 1 4) 11212.
(5.40)
(5.41) (5.42)
Optimal Load Transmission by Flexure
&
$
Q
+
x
YY
(a)
FIG. 35. Solutions for domains with curved boundaries.
245
246
George I. N. Romany and Robin D.Hill
For an elliptic boundary (Fig. 5.352) given by y = + b ( l - x2/a2)1’2,
(5.43)
(5.40) furnishes the elliptic boundary jj
= b[ 1
- js2/a2(1 - b2/2a2)2]”2/2.
(5.44)
For semicircular and diamond-shaped domains the solutions are given in Figs. 35d and 35e-g, respectively. In the former, y, = R sin O/2( 1 + sin O),
X,
cos 8/2( 1 + sin O),
=R
pc = R[ 1 - sin O/2( 1
+ sin O)],
(5.45)
(5.46)
where R is the radius of the domain boundary. In Fig. 35f polar (6, r ) and Cartesian (x, y) coordinates of a general region boundary point and Cartesian coordinates (xH, yH)of the point H,respectively, are
”(
P =2 1 y = (R/2)(1
- cos O),
+
A),
x = (R/2)(sin 0 + tan O),
x = 2(R - y)(Ry - y2)”’/(R
YH = R - XH = R($
VI. Optimal Flexure Fields-Mixed
- 2y),
- I)/@.
(5.47) (5.48) (5.49) (5.50)
Boundary Conditions
A. BRANCHES CONTAINING a
AND
p FIELDS ONLY
Along simply supported boundary segments, three classes of optimal fields termed a, y, and 6 types may occur in the solution. Because of their relative geometrical simplicity, aa- and ap-type branches will be discussed first and branches containing y- or &type fields will be dealt with in Section VI,D. Basic properties of a fields are (i) The field consists of an R+-type region?. t In limiting cases, a fields may degenerate into an S + - or T-type region. The latter permits nonzero moments along the boundary.
Optimal Load Transmission b y Flexure
247
(ii) Both principal moments take on a zero value along the domain boundary. (iii) The curvature is IC, = k along principal lines. It will be seen later that y fields consist of an S--type region and in 6 fields the two principal curvatures take on values of K 1 = k and I C ~= - k along the ~ - k on the interior of the field. domain boundary with I C = uu Branches are bounded by two simply supported edges and afl branches by one simply supported edge and one clamped edge. Their properties are as follows: Property 7. The principal lines of an aa branch may be constructed on the basis of Fig. 36, in which AW = BW, AW I d , , BW 1 d , , AF = BF,
FIG.36. Property 7.
F E h, and ic, = k on AB, where d , and dz are simply supported domain boundaries and h is the branch centerline. The foregoing construction is admissible only if
I f t x(o)
I5
i = 1, 2,
(6.1) where y = f , ( x ) and y = f z ( x ) represent the boundaries d , and d 2 in the coordinates shown in Fig. 36. +fi(O)fi,XX(O)
1 3
Property 8. If (6.1) is satisfied as an inequality then the corresponding part of an ua branch is an R+-type region. If fi(x) = f 2 ( x ) = f ( x ) and (f:x IX=,, = 1, - 1, respectively, then the uu branch degenerates into an S + - and a T-type region. The second principal curvature at domain boundaries is furnished by
+xxx)
K
~
-=k ( f : x + l ; . f i , x x ) I x = O ,
i = 42.
(6.2)
248
George I . N . Romany arid Robin D. Hill
Property 9. The centerline h of a branch may only intersect the domain boundary d at a local maximum of the curvature of d. Positive boundary curvature is defined in Fig. 16. Property 10. If the intercept point W in Fig. 36 is at an equal distance from n domain boundary points with n > 2, then an n-sided S+-typeregion occurs in the solution, which can be constructed on the basis of Fig. 37 (in which n = 3).
FIG.37. Property 10.
Property 1 1 . The principal lines of a/3 branches may be constructed on BW = CB, A W I d,, BC I d , , the basis of Fig. 38, in which A W =
&2
&2
FIG. 38. Property 11.
FW I AB, F E h, and B E b, where d , is a simply supported and d , a clamped domain boundary, h the branch centerline, and b a region boundary. AB and BC, respectively, are principal lines in R + - and R--type regions. The foregoing construction is admissible only if
I
f,’x + f x x
1,
(6.3)
for rc(d2)> 0,
(6.4)
lx=o
and C W c rc(d,),
where y = f ( x ) represents d,, axes x and y are given by FW and FA (Fig. 38), and rc(d2)is the radius of curvature of d, at C .
Optimal Load Transmission b y Flexure
249
Property 12. The centerline of an a/? branch may only intersect the domain boundary at a point where a line of simple support changes to a line of clamped support.
For the construction of junctions of aa, ap, and Property 27.
pp branches, refer to
Property 13. An S--type region occurs at a corner of a domain if the angle between two adjacent sides is
(i) > n/2 for two simply supported edges (Fig. 39a), (ii) > 3 4 4 for one simply supported and one clamped edge (Fig. 39b), and (iii) > n for two clamped edges (Fig. 39c).
(b)
(C)
FIG.39. Optimal flexure fields for corners.
Property 14. Let an ap branch be associated with a clamped-point support or reentrant corner C and a straight simply supported line support AA'
FIG.40. Property 14.
George I. N . Rozvany and Robin D. Hill
250
at a distance a from each other. Then the region boundary is given by (Fig. 40) y i = x i + a2/2.
(6.5)
B. PROOF OF PROPERTIES 7-14 1. Proof of Properties 7 and 8
It follows from Proposition 1 that the line AB in Fig. 36 is a principal line. By substituting = b and = - b into (4.14), we get
r
<
I C = ~
( p - l)k,
I C = ~
(4- l)k,
(6.6) respectively, which yield (6.1)and (6.2)after substitution of (4.15)and equating I x2 I to k for (6.1).The same substitution shows that in the limiting cases considered, x2 = k or x2 = - k, giving S + - and T-type regions.
2. Proof of Property 9
Consider the two adjacent circular domain boundary segments shown in Fig. 41, in which r2 r l . Using the construction in Fig. 36 and evaluating
-=
I’
FIG.41. Proof of Property 9.
(6.1) for fl(x), we get for the point A if it is at an infinitesimal distance to the right from the junction of the circular arcs, f1(o) = rl sin a, fl,,(0)= cot a, f1,,,= - ( I cot’ ~ ) ~ / ’ / r ~
+
- 1/r2 sin3 a. Let the left-hand side of (6.1) be denoted by Iw 1. Then w = (cot’ a - rl/r2 sin2 a), =
(6.71 (6.8)
Optimal Load Transmission b y Flexure
25 1
and for r 1 = r 2 , w = (cos’ a - I)/sin2 ct = - 1.
(6.8a)
Further, for rI > r 2 , w < - 1 and hence (6.1) is violated. If we extend the foregoing argument to the case a -+ 0, -+ 0, then it follows that the curvature cannot decrease as we move away from the intersection of the centerline h and the domain boundary d. 3. Proof of Property 10 For this property the proof is the same as for Property 3 (see Section V,C,7). 4. Proof of Property 11
The construction in Fig. 38 satisfies all requirements of Proposition 8 with = kJZ(BW),
COB
= k(BW),
UA
- U S = k(BW)’/2.
(6.9)
It follows from Proposition 9 that AB is a principal line and the proof of Property 1 implies that CB is a principal line. 5. Proof of Property 12 Since the construction in Fig. 38 is only possible for a point A of a simple supported line and a point C of a clamped line, an intersection of the domain boundary d and the centerline h is at a point where AW -+ 0 and CW + 0, and hence AC 40. 6. Proof of Property 13 This property follows directly from the constructions in Properties 1 and 11, and from the fact that at a corner between two simply supported edges the angular velocity must be zero in all directions. Thus such a corner is kinematically equivalent to a clamped point. 7. Proof of Property 14
In Fig. 40, (BC)2 = X’B
+ (y’B- a)2,
2(yB - a ) = ~ ( B c -) a,
which imply (6.5).
(6.10) (6.11)
252
George I . N . Rozvany and Robin D. Hill
c. c1 AND B FIELDS-EXAMPLES 1. Rectangular Domains with Various Support Conditions Optimal solutions for square and rectangular domains with various support conditions are given in Figs. 42 and 43. For the total "cost '' (volume) of various optimal solutions, the reader is referred to an earlier paper (Rozvany and Adidam, 1972~). Considering a rectangular domain having changes in support conditions at points other than corners, the solution takes the form shown in Fig. 44.
FIG.42. Optimal flexure fields for square domains.
25 3
Optimal Load Transmission by Flexure
'I
L/4 ' L/4
I
L/4
I
L/4
q(td
a L
a L
la,Al
4 (e)
FIG.43(a)-(f). Optimal flexure fields for rectangular domains.
254
George I . N . Rozvany and Robin D. Hill
Optimal Load Transmission by Flexure
255
FIG.44. Rectangular domain with changes in support conditions along the edges.
2. Simply Supported Triangular Domains Figure 45 shows the solution for simply supported triangular boundaries. The solution for three acute corners (Fig. 45a) is due to Lowe and Melchers (1972). If one of the corners forms an obtuse angle then the S+-typeregion splits into two parts separated by an R+-type fan (Fig. 45b). 3. Simply Supported Quadrilateral Domains Figures 46-50 (on pp. 257-265) show the 32 different topographies that the optimal solution may take on for simply supported quadrilateral domains. It is rather surprising that for such a simple class of boundary conditions of constant topography (four sides, four corners), the solution may have so many different configurations. Properties of various topographies are listed in Table 2 (on p. 266). Under the classification “type”, 0 refer to obtuse and N to nonobtuse corners (including corners enclosing n/2),and the corners are listed in a cyclic sequence. S S Connections refer to an R+-type region having principal lines in a constant direction and connecting two opposite sides of the domain (see Fig. 46d, for example). Sides are denoted by stating the type of corners at their ends, and subscripts are used in identifying corners only if it is necessary for distinguishing between two topographies.
256
George I. N. Romany and Robin D. Hill
(b) FIG.45. Triangular domains with simple supports.
CS Connections take place between a corner and a side in the form of an S--type region at the corner and an R+-type fan at the side. Two such connections may be observed in Fig. 46c. CC Connections link two corners via two S--type regions and a T-type region, as at the top of Fig. 46h. Lowe and Melchers (1973% Fig. 2) proposed a solution for quadrilateral domains that fails to satisfy the inequality condition in (4.1) at obtuse corners. The above solution would lead to the erroneous conclusion that the two topographies given in Fig. 46a,b can be used for all convex quadrilateral domains. Although the region configurations given in Figs. 46-50 are often valid for some more general cases, many of the actual examples given.belong to a specific class of boundary shapes. For example, Figs. 46c,d and f-h show
Optimal Load Transmission b y Flexure
(a)
(b)
a =c
FIG.&(a)-(d).
Quadrilateral domains with simple supports.
257
258
George I . N . Romany and Robin D. Hill
FIG.46(e)-(h)
Optimal Load Transmission by Flexure
0 -
FIG.47(a)-(d). Quadrilateral domains with simple supports.
259
260
George I . N . Romany and Robin D . Hill
(i)
FIG.47(e)-(i)
Optimal Load Transmission by Flexure
(C)
FIG.48(ab(c). Quadrilateral domains with simple supports.
26 1
262
George I . N . Romany and Robin D . Hill
t 5 FIG.48(d)-(f)
Optimal Load Transmission by Flexure
263
(C) /
FIG.49(a)-(c).
x
Quadrilateraldomains with simple supports.
264
George I . N . Romany and Robin D. Hill
v
FIG.49(d)-(f)
Optimal Load Transmission by Flexure
FIG.50. Quadrilateral domains with simple supports.
265
TABLE 2 CLASSIFICATIONOF THE TOPOGRAPHIES OF SIMPLY SUPPORTED QUADRILATERAL DOMAINS
Type
1
SS Connection
NNNN
NN-NN
I
I
ONNN
ON-NN
1
ON-ON
CC Connection
00-NN
0-0
0-0
CS Con-
O-ON
nection
Ol-NN O,-NN
Ol-NN 02-NN
0-NN
22;
02-01N
Ol-NN 02-NN
47b
47a
47c
46c
Figure 46b
47i
47h
48c
I
number
46d
46h
46f
I
ON-ON
1
47d
I
ONON S S Connection
46g
47g
OOON
OO-ON
146el
47f
1
48b
I
48a
I
47e
49a
49c
48e
49b
50a
50b
48f
49
49e
49f
48d
5Oc
Optimal Load Transmission by Flexure
267
solutions for symmetric trapezoidal domains. The four parameters that govern the choice of optimal topography are a, c, 1; and g, where
f = c - (c2/2 + a2/16)'/2, g = a tan(n/4
+ a)/4.
(6.12) (6.13)
The value off is based on the construction given in Fig. 38 with f i C B =
f i BW = AW and the additional requirement that Wand A fall on the axis of symmetry (Fig. 46h). Point B in Fig. 46h represents the corner of S - - and S+-type regions and f is the distance of B from the boundary segment E. g represents the distance of point L in Fig. 46g from E. Figures 46f, g, and h give solutions for domains with c > a and for f = g, f> g, and f< g, respectively. Figures 46c and d correspond to cases with a = c and a > c. 4. Domain Bounded by Two Simply Supported Circular Arcs (Fig. 51a)
Before considering the above problem, it is convenient to determine first the optimal solution for a simply supported circular domain that is clamped at one boundary point only (point C in Fig. 51b). Using the construction given in Fig. 38, the three side lengths of the triangle CWZ in Fig. 51b take on the values 2a, R - $a, and R, respectively. Then the length a can be easily expressed using the cosine law: a = R(2 cos a -
fi),
(6.14)
which gives the region boundary in Fig. 51b in polar coordinates. Parts of the R + / S region boundary and of the displacement fields shown in Fig. 51a will be used in the optimal solutions to be considered next. Returning to the problem in Fig. 51% the optimal region topography depends on the relative proportions of the domain, which can be defined uniquely in terms of the angle 1in Fig. 51a. (i) 1 2 45". It follows directly from Property 13 that the simplest optimal solution shown in Fig. 52a is valid only if 12 45". However, it is still necessary to show that (6.1)is satisfied for all points of the boundary. Adopting the coordinates shown in Fig. 52a, the equation of the boundary becomes & y = & (R2 - x')''~ T A.
(6.15)
It can then be checked easily that (6.15) satisfies (6.1)even for the limiting case 1 = 4 4 , A = R/$, Q Ix I (ii) 19.47"< 1 < 45". For this range of values, the optimal solution is shown in Fig. 52b, where the curved boundary is given by (6.14) and the
RIG.
268
George I. N. Romany and Robin D. Hill P
R
4 (b)
FIG.51. Solution for simple supports along two simply supported circular arcs.
adjacent R+-type region contains the same displacement field as the corresponding portion of the R+-type region in Fig. 51b. (iii) 1= 19.47'. This is a special case when the solution contains only S + - and S--type regions and R+-type fans. On the basis of Fig. 38, R cos 1 =
Jz ~
( -1sin A),
(6.16)
which gives 1 = arc sin(+) = 19.47'. (iv) 0 < 1 19.47' and -90" I 1< 0. For these ranges of values, the optimal solutions are shown in Fig. 52d,e, respectively. 1= 0 is excluded from the range of values considered because at that value the boundary reduces to a simply supported circle and hence the fixity at the corners is temporarily removed.
-=
Optimal Load Transmission by Flexure
t'
n
A>- 4
FIG
9
(a) 19.47"<'A
45'
269
George I. N. Romany and Robin D. Hill
270
l-
a
a
a
1-
1-
W
a12
4
W
(b) FIG. 53. Simply supported T-shaped domain.
5. T-Shaped Domains with Simple Supports
Optimal solutions for T-shaped domains are given in Fig. 53. While in Fig. 53a the side lengths are such that b c a, Fig. 53b shows the changes in the top part of the solution if the above inequality is reversed. 6. Solutions Consisting of a Single S + - or T-type region In the problems that follow, the functions y = k f ( x )representing symmetrically located simple supports give 1 or - 1 for (f.: + ffxx) and thus by Property 8 the solution degenerates into an S + - or T-type region (see Fig. 54). (a) Straight edges at right angles (T-type; Fig. 54a) y = +x.
(6.17)
Optimal Load Transmission b y Flexure
27 1
FIG.54. Solutions consisting of a single region.
(b) Circular boundary (S+-type; Fig. 54b) y2 = r’ - ~
2
.
(6.18)
(c) Hyperbolic boundaries (T-type; Fig. 54c,d) y2 = x 2 - a’
or
y 2 = x 2 + a’.
(6.19)
7 . Domains Bounded b y Straight-Line Segments
Property 15. If the boundary of a domain consists of only straight clamped and simply supported line segments then all region boundaries comprise straight line segments, parabolas, or hyperbolas. Proof of Property 15. On the basis of Properties 1, 7, and 11, region boundaries associated with two straight domain boundary segments are themselves straight. Region boundaries associated with a reentrant corner
FIG.55. Example of an optimal flexure field for mixed boundary conditions.
272
George I . N . Rozvany and Robin D. Hill
and a clamped or simply supported line segment, respectively, are parabolas or hyperbolas (by Properties 6 and 14). The foregoing property may be observed in Fig. 55, in which curves PQ and WZ, for example, are hyperbolic and parabolic, respectively.
D. y AND 6 FIELDS 1. Introductory Remarks on Modes of Load Transmission to Simply Supported Boundaries
If the domain D of a grillage is located between two simply supported boundaries (Fig. 56a) then the most obvious layout for transmitting downward loads appears to be a series of simply supported beams in positive
FIG.56. Modes of load transmission to simple supports.
bending. However, the foregoing arrangement, termed an aa branch, is not always statically admissible; moreover, limitation (6.1) indicates that aa branches are often not optimal. Figure 56b shows a problem in which the domain is located on the convex side of a single circular simply supported boundary segment and the remainder of the boundary is unsupported. Then the only statically admissible load transmission pattern is the one indicated in which a circular beam is placed over the simple supports and radial beams in negative moment carry the load. Since simple supports are capable of transmitting any set of vertical forces, the moment along the circular beam may vary according to the requirement for balancing the moments in the radial beams. The variation of the circumferential moment may include finite steps because concentrated couples may be resisted in the direction of the simple support. The type of load transmission pattern shown in Fig. 56b is termed a y field. Another mode of load transmission along simply supported boundaries is shown in Fig. 5612, where a point load P is transmitted to a clamped point or corner of a simply supported boundary by a series of beams that are subjected alternately to positive and negative moments. If the plane of the resultant of the moments in the two beams intersecting at a simple support is tangential to the line of support, then such an arrangement is statically
Optimal Load Transmission by Flexure
273
admissible. Owing to kinematic optimality conditions to be discussed subsequently, however, the length of the beams in positive bending can only be infinitesimal, and therefore the entire load is transmitted from the interior of the domain to the boundary by the beams in negative bending. The resulting load transmission pattern is termed a 6 field. It will be shown subsequently that the ranges of validity of a, y, and 6 fields in optimal solutions complement each other for various types of simply supported boundaries.
2. Properties of y Fields Simple examples of y fields are the inner regions of axisymmetric flexure fields in Figs. 13c,gj, and 14c,&j. It can be seen that y fields have the following properties: (i) They consist of an S--type region. (ii) Along the boundary, the angular velocity in the normal direction is
w = kr,
(6.20)
where r is the radius of curvature of the boundary. (iii) y Fields may only occur along simply supported boundaries of constant radius. Equation (6.20)can be obtained by equating K@ to -k in (4.55). In the foregoing examples in Figs. 13 and 14, both Mr and Me may be nonzero in the S--type regions, and therefore an infinite number of alternate moment fields give the same optimal cost. However, if the simple support forms an incomplete circle only (Fig. 57) then the static interpretation of a y field becomes more difficult. The moment fields in Fig. 57b,c satisfy the equilibrium equation (rMr),rr
‘1-
n
(6.21)
- Me,, = -Pr,
rM,
(b) Me
+I+ a
(C)
FIG. 57. Statical interpretation of y fields.
(D
George I . N . Rozvany and Robin D. Hill
274
but the circumferential moments of intensity A cannot be balanced at the free edges of the plate. This means that for a given value of rM, = Aa at the simple support, a + 0 must be adopted and then M, is converted into an impulse function over the simple support. At the ends of the latter, concentrated vertical couples in a plane tangential to the line of support equilibrate the circumferential moment-impulse. The physical manifestation of this solution is a concentrated beam in grillages or concentrated reinforcing fiber in fiber-reinforced plates (Fig. 57e) along the line of support. The concentrated fiber may be “anchored” at the ends of the support (Fig. 57f). The static interpretation of y fields for point loads requires some further explanation. Figure 58a shows a point load on an annular domain with an
P
_II
I
I
(b)
(C)
(d) FIG.58. A y field for a single point load.
inner simple support only. The optimal moment field consists of a concentrated radial moment M , along P A and an M , impulse of both infinitesimal length and width over the line support at A. The correct “cost” of this solution may be derived from either static (4.3) or kinematic (4.5) considerations. Equation (4.5) gives (Fig. 58c) Omin= fi = P k ( R , L + C/2).
(6.22)
In deriving the moment volume by a static method (4.3),it is first assumed that P is distributed over a circular line segment in circumferentialdirection
Optimal Load Transmission by Flexure
215
corresponding to a central angle 8 (Fig. 58d). Then the radial moment at the simple support takes on the value (6.23)
(6.24) in which PL/8 corresponds to the area Aa of the M , diagram in Fig. 57a. Hence the integral of the circumferential moments M , becomes MOrdr = (8R1)(PL/B) = PLR,.
(6.25)
Since the above result is independent of the value of 8, it still holds for 0 + 0. The radial moment volume (Fig. 58b) is clearly PL2/2, giving the cost value in (6.22) after multiplication by k. 3. Properties of 6 Fields It has already been indicated that in 6 fields only negative moments may occur on the interior of the field but there are both positive and negative principal moments along the simply supported boundary. This means that by (4.1) 6 fields are R--type regions that must fulfill the following: Jul I s k ,
x t = -k, =
-k,
on Int D,
(6.26)
u1 = k, on Bd D.
(6.27)
Using the foregoing relations, the following properties of 6 fields may be obtained. Property 16. Considering a circular boundary segment having the radius of curvature -r, let the origin of polar coordinates be at the center of the circular boundary. Then for the boundary of a 6 field, the angular velocity il, in the normal direction and the angle a between the principal lines and the boundary are given by iii = kr cos 2a,
(6.28)
a=c
(6.29)
- 8/2,
where C is a constant and 8 the circumferential coordinate.
276
George I . N . Romany and Robin D . Hill
Since the angular velocity along and in the direction of the boundary is zero, it follows from the above property that the angular velocity w in the direction of the principal lines is w = kr cos 2a sin a,
(6.30)
and in a direction normal to a principal line w2 = kr cos 2a cos a.
(6.31)
If 8 = 0 corresponds to the end of a clamped segment (Fig. 59) then the zero angular velocity requirement at 8 = 0 gives C = 4 4 = 45". A complete 6 field is shown in Fig. 59% where the domain has only an internal support along a circular line that is simply supported along AD and clamped along AB.
FIG.59(a). A S field for a circular boundary.
Optimal Load Transmission by Flexure
277
(C)
FIG.59(b) and (c)
Figure 59b shows the centerline (AGC),the second contour line ( A E K C ) with u = 0, and the envelope (JEGC) for the velocity field in Fig. 59a. All three lines and the boundary have a common point and the same tangent at C. The centerline is the locus of the maximum velocity along each principal line if the latter are extended with a curvature rate K = - k beyond the boundary. The distance (e.g., FM or G H ) measured along a principal line, between the boundary and the centerline equals w/k and thus shows graph-
278
George I . N . Romany and Robin D . Hill
ically the variation of the angular velocity w along the boundary.? The distance of the envelope from the boundary (measured along principal lines) is (2r sin a)/3. It also follows from Property 16 that the angular velocity a along and in a direction normal to the boundary may only vary between the limits
0 Ia Ikr.
(6.32)
Property 17. Along a simply supported boundary of varying curvature, (6.28) still holds but (6.29) is replaced by the differential equation (Fig. 60)
da/dO = -$
+ (dr/dO)(1/2r)cot 2a,
(6.33)
FIG. 60. Derivation of properties of 6 fields.
t The second contour line with u = 0 has a cusp at point E, which is also a point of the envelope. This can be explained by the fact that the segment EKC of the second contour line does not lie on the same velocity surface as our 6 field but on its extension beyond the envelope. Looking again at Fig. 10%we can see that each principal line of an R+-field is really a half-line (e.g, MN).If we extend them beyond the envelope, then we get another R+-field that is, in general, different from the one considered originally. The former will be termed the conjugate of the latter. Returning now to Fig. 59b, we can see that the segment EKC of the second contour line is on the conjugate field of our 6 field. It is worth pointing out that Proposition 6 is still valid for the principality of a basic line ofan R+-field even when the centerline is contained in the conju_gate field. Theonly difference is shown in Fig. 59c, where F F is a centerline contained in an R+-field and FF' is a centerline in another problem in which the latter is contained in the conjugate field. In Fig. 59b, the principal line EF along which the second contour has a common point with the envelope is associated with aF = 35.264 and OF = 19.471, and the principal line GH containing the intersection of the centerline and the envelope is characterized by aH = 24.094 and OH = 41.810.
Optimal Load Transmission by Flexure
279
where 8 is the angle between the tangent to the boundary at a point and a reference direction and r is the radius of curvature of the boundary at the same point. 4. Proof of Properties 16 and 17 Considering the extension of a 6 field beyond the boundary (Fig. 60),let the origin of Cartesian coordinates (x, y ) be at a distance o / k along a principal line from the boundary. Owing to the curvature rate - k in y direction, the point (0,O)will then correspond to the maximum velocity u along the principal line considered. The value of w given in (6.30) corresponds to (6.33a) where y = f (x) represents the simply supported boundary. Since the latter is a contour line, we can use (4.14) and (4.15) with = b for calculating rcl at A . t Setting -f (0) equal to the right-hand side of (6.33a), we get
<
~ 1 = ( ~ - l ) k = { f , 2 *l+(l-f:x)-l}k=k, +
(6.34)
which satisfies condition (6.27). It is still necessary to show that AF in Fig. 60 is a principal line. For r = const, Ar = 0 and a in Fig. 60 becomes a = r Ad cos a
+ 0(AO2)+ O(Aa2),
(6.35)
and Aw = r cos 2(a + Aa) sin(a + Aa) - r cos 2a sin a =r
Aa(cos a cos 2a - 2 sin a sin 2a) + O(Aa2).
(6.36)
By (4.131, (a - Aw)
+ O(da2)+ O(Ad2)= (Ad - A a ) w 2 ,
(6.37)
where w2 is furnished by (6.31). Substituting (6.35), (6.36), and (6.31) into (6.37) and neglecting terms of higher order, we get Aa = Ad(cos a cos 2a - cos a)/2 sin a sin 2a = -Ad/2,
(6.38)
which implies (6.29).
t The sign of the right-hand side of (4.14) must be changed because here we are dealing with R - - and not an R+-type region.
280
George I . N . Romany and Robin D. Hill
When r is a variable quantity, then after neglecting terms of higher order, (6.36) changes to
A o = Ar sin a cos 2a + r Aa(cos a cos 2a
- 2 sin a sin 2a).
(6.39)
Equations (6.39) and (6.37) give
Au = -$A0
+ (Ar/2r) cot 2a,
which implies (6.33). 5 . 6 Fields-Examples (a) Circular Boundary with a Clamped Point. Figure 61% in which the edge is clamped at point A, shows a solution that consists of one fl, two 6, and one y field. A similar solution would be obtained if the clamped point in Fig. 61a were replaced with an obtuse corner (Fig. 61b). (b) Domain Bounded by Two Circular Arcs. Figure 62 indicates an optimal region configuration containing one aa branch and two 6 fields separated by a T-type region. The limiting angle E for the aa branch is given
(bl FIG.61. Examples involving 6 fields.
Optimal Load Transmission by Flexure
28 1
FIG.62. Example of adjacent aa and SS branches.
by (6.1) in Property 7, in which f x
= tan
E,
f = L(1 - cos E ) ,
fxx
= (1
+ tan’ E ) ~ / ’ / L(6.40) .
Then (6.1) reduces to (tan E)’
+ (1 - cos E)( 1 + tan’
E)~/’I
1,
(6.41)
or E
s arc c0s(O.5)’/~= 37.46731.
(6.42)
Beyond this angle, limitation (6.1) makes an aa branch inadmissible. Since the curvature rate along AB in Fig. 62 is k, the angular velocity w at A in y direction is w = kb = kyl - cos E ) .
(6.43)
Combining (6.41) as an equality and (6.43) we get w = kL cos E cos 2 ~ .
(6.44)
This means that the cla branch becomes inadmissible exactly at an angle E at which the slope at the boundary in the direction of the principal curvature
282
George I . N . Romany and Robin D . Hill
rate I C ~is the same as the value required for 6 fields [see (6.31)]. Th’is agreement provides an independent check on the validity of the theories presented because the relevant conditions have been derived from two entirely different optimal velocity fields, namely, a and 6 fields. 6. The Complementary Nature of a, y, and 6 Fields
The range of validity of a and 6 fields also complement each other for other boundary shapes. If we replace the circular arcs in Fig. 62 with boundaries given by y = kf (x) and if E is the angle between the tangent to the boundary and the axis x, then by (6.1) the upper limit for the angle E for aa branches is given by tan2 E + 2b( 1 + tan’ ~ ) ~ / ’=/ r1,
(6.45)
where r is the radius of boundary curvature and b =f(O). At an arbitrary boundary point, the maximum kinematically admissible angular velocity o in the y direction is w = kb and by (6.31) a 6 field requires w = kr cos 2~ cos E in the same direction. This means that the lower limit on the angle where a 66 branch can take over is given by b = r cos 2~ cos E.
(6.46)
It can be shown readily that (6.45) and (6.46) yield the same value for E. Similarly, the maximum angular velocity for which a 6 field can be constructed is given by (6.32), and at the same angular velocity value the 6 field can change into a y field. This can be seen in both examples in Figs. 61 and 62.
E. BRANCHESCONTAINING y AND 6 FIELDS Property 18. ay Branches are bounded by a simply supported circular boundary segment and a simply supported boundary segment of general shape. The construction of ay branches is given in Fig. 63, where d’ = (e’ + r2)/2. (6.47) For the particular case of a circular and a straight boundary the solution is given in Fig. 64, where the region boundary is represented by y’ = x’ + (C- r2)/2. (6.48) The limitations of the validity of Property 18 are r I d,
(6.49)
and if the other boundary is represented by y =f (x) then it must satisfy (6.1), for which the axes x and y coincide with FW and FB in Fig. 63.
Optimal Load Transmission by Flexure
28 3
FIG.63. Construction of ay branches.
A
>
r
x
,
B
FIG.64. Example of an ay branch.
Property 19. y/3 Branches are bounded by a circular simple support and a clamped support of general shape. Their construction is shown in Fig. 65, where f2
= g2
+ r2/2,
(6.50)
and A and B are contained in the region boundaries. Property 20. yy Branches are bounded by two circular simple supported lines. Their construction is shown in Fig. 66, where 2b2 - r: = 2c2 - r22 9 and A and B are contained in the region boundaries.
(6.51)
284
George I . N . Rozvany and Robin D. Hill
FIG.65. Construction of B y branches.
FIG.66. Construction of yy branches.
The foregoing construction gives identical results to the one shown in Fig. 67a, where (6.52) L , , ~= ~ / f 4 (r: - r3)/2L. This construction is valid only if
L12 rl,
L2 2 r2 .
(6.53)
For rl = r2 = p, the same condition reduces to L 2 4p.
(6.54)
The limiting case with L = 4p is shown in Fig. 67b.
(a) FIG.67. Examples of y y branches.
(b)
Optimal Load Transmission by Flexure
28 5
Proof of Properties 18-20. It can be checked readily that the constructions in these properties satisfy the requirements of Proposition 8 and thus ensure slope continuity along the region boundaries. Further, Proposition 9 implies that line segment AB is a principal line in Figs. 63, 65, and 66. Property 21. A 6 field associated with a circular simply supported boundary with a clamped point D may be constructed on the basis of Fig. 68, where EH is a principal line.
tH
FIG.68. Construction of 6 fields.
Further, the angular velocities along EH are the same in all directions as for a y field having the circular boundary with its center at G and radius GE. Proof of Property 21. By (6.29)with C = 44, a = (n/2 - q / 2 ,
(6.55)
and then by (6.28) the angular velocity normal to the boundary at E is (see Fig. 68) c-3 = kr cos 2a = kr sin 8.
(6.56)
In the construction shown, k(EG) = k(EF) and the latter clearly equals the value in (6.56).Hence a y field with boundary radius EG and center G would give the same angular velocity as a in (6.56)at E and in a direction normal to the boundary. The angular velocity in the direction of the boundary is zero for both fields. Since EH is a line of principal curvature rate for both the S field and the “equivalent” y field, the angular velocity in a direction normal to this line is constant and the curvature rate along EH is K = - k for both fields. It follows that the angular velocity along EH is the same in all directions for both the 6 field considered and the equivalent y field.
286
George I . N . Romany and Robin D . Hill
A
I
i n'
1
(b) FIG.69. Construction of 66 branches.
Optimal Load Transmission by Flexure
287
Property 22. 66 Branches associated with a single simply supported circular boundary and clamped points (Fig. 69a) or corner points (Fig. 69b) can be constructed on the basis of Fig. 69%where lines of negative principal curvature are E A and E'A' and the line of positive principal curvature is AA'. Centers of the equivalent y fields are G and G . Then (6.52) with rl = r2 yields AA' = i G G = a/2.
(6.57)
The asymptotes of the region boundary can be determined by considering the case when the lines of principal curvature D A become parallel to each other (Fig. 70a). At these values of 0 and a, 2a = fn - 0,
a
= a n -40 = 8 - B,
(6.58)
and hence
0 = in
+ $3,
(6.59)
which gives (Fig. 70a) a = r[ 1 - sin(4n
+ +COS(& /I)] - fp),
(6.60) where the distance of the asymptote from the centerline is a/2. For p = 0, a = r(1- sin in) cos in = rJ5/4.
(6.61)
The region boundary and directions of nonzero moment for p = 0 are shown in Fig. 70b, where the asymptotes of the region boundary are shown by the dashed line. The region boundary in Fig. 70b can be derived easily from Property 22 (Rozvany and Hill, 1973): x2 = r2w3(2- w)/4,
y = rw(2w2 - 3w
+ 4)/(2 - 4w),
(6.62)
with w = 1 - cos 2a = 1 - sin 8. Properties 23-26. The construction of (i) 66 branches associated with two different circular boundaries and (ii) ad, PS, and y6 branches, is shown in Figs. 71 and 72, respectively. Outline of Proof; Properties 22-26. In the constructions in Figs. 69, 71, and 72, first the center of the equivalent y field is found on the basis of Property 21. Then the region boundaries are constructed by making use of Properties 18-20. Property 27. The construction of the junction of branches containing a,
/?,y, and 6 fields is shown in Fig. 73, where K A = A W = a, EB = BW = b, C W = c, FD = D W = d, K G = g = p1 cos 2a, 2a2 - g2 = 2b2 = c2 = 2d2 - p2.
288
George I . N . Rozvany and Robin D. Hill
FIG.70. Example of a SS branch.
Optimal Load Transmission by Flexure
FIG.71. Construction of SS branches for two circular arcs.
289
290
George I . N.Romany and Robin D . Hill
FIG.72. Construction of a6, b6, and yS branches.
Optimal Load Transmission by Flexure
29 1
FIG.73. Construction of junctions.
Outline of Proof, Property 27. It can be checked easily that triangles ABW, ADW, DCW, and BCW satisfy all requirements of Proposition 8. Therefore, the proof given for Property 3 is also valid for the above triangular segments of the junction.
Remark. Figure 73 also represents a concise summary of the construction of all ten types of branches if we consider, in turn, two boundary points (out of E, C,H,and G) at a time. A unified theory for the four types of fields was outlined by Romany (1975b).
F. EXAMPLES OF OTHER TYPES OF INTERFIELD BOUNDARIES In Fig. 61, a /I and a y field were separated by two 6 fields, and in Fig. 62 a 66 branch was between an aa branch and two y fields. In Fig. 74a, a 6 field is adjoined by two y fields and Fig. 74b shows an example in which an act branch changes into a y y branch. Both solutions can be constructed readily on the basis of the relevant properties. In Fig. 74c,the domain has three free edges and an inner simple support of varying curvature. A y field is located
292
George I . N . Romany and Robin D . Hill
(a)
0
FIG.74. Other types of interfield boundaries.
at the point A of maximum boundary curvature and at the same point a = 0 for the adjacent 6 field. If the boundary curvature were constant, an increase in 8 would require a decrease in a (by 6.29), which would not be possible at A because a is already zero. However, relation (6.33) can give a positive increment of a for an increase in 8 and thus the beam arrangement shown by the dashed line can transmit the load from the 6 field to the y field.?
VII. Optimal Flexure Fields-Further Developments In this section, only a few concise conclusions on various other aspects of optimal flexure fields are presented.
A. THEEFFECT OF COMBINED BENDINGAND TORSION A typical elliptic contour line $(M,, MXy) = const for the specific cost of a beam in combined bending (M,) and torsion (MXY) is shown in Fig. 75a (e.g., Hodge, 1959) and the corresponding optimal strain rates furnished by (2.11) are indicated in Fig. 75b, where K , and K,, are the curvature rate and rate of twist, respectively. t In Fig. 7% the beam in positive bending has a finite length. In the actual optimal solution, the length of such beams tends to zero.
Optimal Load Transmission by Flexure
293
FIG.75. Cost contour and cost gradient surface for bending and torsion.
Clearly, for beams in the direction of principal moments (M, # 0, M,, = 0), the curvature rate requirement K, = k given by point A is satisfied by all solutions in this article. In other directions, M, = 0, M,, = 0, and thus (2.11) admits any strain rate representing a point on or inside the “cost gradient surface” in Fig. 75b. The maximum twist occurs in a T-type region and its magnitude is I K,, 1 = k (point B in Fig.’ 75b), which is well inside the curve in Fig. 75b. It can therefore be concluded that the optimal grillage consists of beams free of torsion if a solution satisfying (4.1) can be found.
B.
THEEFFECT OF COMBINED BENDINGA N D
SHEAR
It has been demonstrated (Watson, 1973) that in the case of grillages consisting of long-span trusses, the specific cost function in (2.15) gives a realistic estimate of the specific structural weight and thus the optimal strain rates in (2.16) may be used for deriving solutions of least weight. Solutions for annular simply supported grillages under axisymmetric loads have been determined by Rozvany and Watson. Making use of (2.16) and adopting values of R and 1.0 for the radius of the inner and outer supports, the range of validity of three types of optimal topographies is shown in Fig. 76 (Watson, 1973), where solution B is valid for the range of values 3R 2 1 - 2kl(l - 2v)/k 2 R,
(7.1)
where v = (1 - [(R’
+ R + 1)/3]”’}/(1
- R),
(7.2)
294
George I . N . Romany and Robin D . Hill
t
I
@
iI
0.4
-Mr 03
i-
0.2-
M0.1-
r
A
i 0.01
0.1
1 .o
10
FIG.76. Optimal solutions for combined bending and shear in an annular grillage.
and v( 1 - R ) is the distance of zero shear from the outer edge. For solution A, the radial length of positive moment is L = {[l
+ 2k,(l - R)(1 - 2v)/k - R2]/2}”*,
(7.3)
and for solution C, v = [2 - k( 1
+ R)/k,]/4.
(7.4)
C. SOLUTIONS FOR INTERNAL SUPPORTS
If a flexure field is continuous over a line of simple support that is on the interior of the domain D, then the optimal velocity field u is dependent on the nonnegative load P( ) > 0, and it is necessary to generate a set of optimal solutions in which the lines of zero moment are variable, depending on the load distribution. Such a set of optimal solutions can be constructed on the basis of Fig. 77, where line A L is an exterior clamped support, ER an exterior simple support, and H M an interior simple support. In the construction shown, the line of zero deflection J N a n be chosen arbitrarily, subject to the condition 1 tc2 I I k at C and D, which can be checked readily by making use of (4.14).For the latter, the contour lines are HA4 and J N . In Fig. 77, GH = GJ = r,
GC = CF = a,
FB = BA = b,
GD = D K = c,
K E = d, 2a2 = r2 + 2b2, 2c2 = d2 + r2, K E I ER, GH I H M , GJ I J N , BA I AL,
(7.5)
Optimal Load Transmission by Flexure
295
and lines AB, BC, CD, and DE are principal lines having the sign of curvature rates indicated. It can be shown readily that the foregoing construction satisfies slope
FIG.77. Construction of solutions for internal supports.
continuity and principality conditions, because along CH and J D the angular velocities and curvature rates are the same as for the “equivalent” y field having the circular boundary shown in Fig. 77. Then Properties 18 and 19 ensure optimality for the construction indicated.
296
George I . N . Rozvany and Robin D . Hill
D. COMBINED UPWARD AND DOWNWARD LOADS Up to this point, it has been assumed that the loads are nonnegative (i.e., downward) throughout D, P( ) 2 0. If the foregoing restriction is relaxed, then the solution takes on a different form. For example, Fig. 78a shows an optimal solution for a rectangular simply supported domain on which the load is partly upward and partly downward, with the additional property P(X,
Y ) = - P( - x, Y ) .
(7.6)
E. OPTIMAL GRILLAGES FOR ALTERNATE LOADS Let the rectangular grillage in Fig. 78 now be subjected to two alternate loads P, and P, such that P,(-X, Y ) = P,(X, y ) 2 0, P,(x, y) = 0 for x > 0,
P,(x, y) = O
for x < O .
(7.7) (7.8)
Using the Prager-Nagtegaal superposition principle (Section II,K), P+ is symmetric and P- is skew symmetric in this example, and hence the corresponding optimal velocity fields are shown in Fig. 78b,a, respectively. Considering, for example, the alternate loading conditions of (1) two point loads P at points A and (2) two point loads P at points B in Fig. 78c,d, P+ consists of P/2 at both points A and B, and P- consists of P/2 at points A and -P/2 at points B. The corresponding optimal grillages are given in Fig. 78c,d, respectively. The moment capacities may be derived by simple statics and the final solution is obtained by superimposing the beam layouts in the foregoing diagrams. F. SOLUTIONS FOR OTHER PROBLEMS INVOLVING OPTIMAL FLEXURE FIELDS The optimal solution has also been determined for a grillage with a reaction force of nonzero cost (Rozvany, 1974a) and for axially symmetric fiberreinforced plates subjected to several alternate load conditions (Charrett and Rozvany, 1972a), having a specified minimum specific fiber volume (Charrett and Rozvany, 1972b) and having specified factors of safety against local yield and ultimate collapse (Rozvany and Cohn, 1970).
RESULTSSUPERSEDED BY THE PRESENT THEORY G. PUBLISHED It is now a point of purely historical interest that several authors who have made significant contributions to this field have also published, in highly reputable journals, results that are inconsistent with the present state of our understanding of optimal flexure fields.
Optimal Load Transmission by Flexure
297
t’
(d)
FIG.78. Solution for alternate loads.
Thakkar and Sridhar Rao (1970), for example, claimed to have obtained the optimal solution for clamped square domains, which has turned out to give about 8% higher cost than the optimal solution in Fig. 42b. Morley (1966) has even expressed doubts about the existence of a solution satisfying (4.1) for the same problem. Lowe and Melchers (1972, 1973a) went even further and claimed that, considering the minimization of the moment volume for plates, there are “in general ” discontinuities in the slope (angular velocity) across region boundaries; they replaced the second part of (4.1) with K~ #
k,
for M i = 0,
(7.9)
“implying that discontinuities in the slope of the deflected plate could exist.” Whereas it is unclear as to why xi = k was excluded from the admissible range of values in (7.9), a major part of a Ph.D. thesis from Cambridge (Melchers, 1971) was devoted to the justification of discontinuities of
298
George I . N . Rozvany and Robin D . Hill
displacements (velocities) in minimum moment volume problems. The issue is of considerable importance, because the claim by Lowe and Melchers, in effect, challenges the validity of theories by Prager and Shield (1967), Morley (1966), Masur (1970), Save (1972), Mroz (1972), and Charrett and Rozvany (1972a). However, the following facts seem to disprove the existence of kinematic discontinuities in optimal flexure fields: (i) The formal proof of slope discontinuities (Melchers, 1971)appears to be based on a misapplication of the Weierstrass-Erdmann corner condition. (ii) All rigorously derived theories listed above require slope continuity. (iii) Optimal solutions satisfying slope continuity are now available for almost all possible boundary conditions and loading. (iv) Slope discontinuous solutions proposed by Lowe and Melchers (1973a,b) have been shown to be nonoptimal (Morley, 1972; Rozvany, 1973d, 1974f)by obtaining statically admissible solutions of a lower moment volume. An example of such solutions is a simply obtuse corner with a point load (Fig. 79a) for which the solution satisfying (4.1) is defined kinematically in Fig. 79b and the solution by Lowe and Melchers is given in Fig. 79c,d. In the
.-
FIG.79. Optimal and nonoptirnal solutions for obtuse corners.
latter, the velocity field is u = k(L2y2 - x2)/2 with K, = - I 2 , which clearly violates (4.1) for obtuse corners. Static interpretations of the solution in Fig. 79b are given in Fig. 79e,f. It is to be noted that for I = 5 in Fig. 79a, the solution (Fig. 79c) violating (4.1) gives 25 times as high a cost as the one (Fig. 79b) that satisfies the same condition. In a very recent paper (Melchers, 1975b), it is proposed that for a continuous beam with two unequal spans and three loading conditions (load on either or both spans) the optimal solution always has a zero moment capacity over the central support. This solution again disregards kinematic optimality conditions, such as (2.44)-(2.46) but solutions satisfying the foregoing conditions have been found to give a lower cost.
Optimal Load Transmission by Flexure
299
H. THEPROBLEM OF FREE EDGES If a free (unsupported) edge of a grillage is in a direction that does not coincide with optimal principal lines for the given support condition, then there appears to be some difficulty in satisfying (4.1). A current investigation seems to be in the process of solving this problem by employing the load transmission pattern in Fig. 80, in which each beam
FIG.80. Proposed solution for free edges.
in positive bending is supported by a beam of infinitesimal length in negative bending, which transmits the load to the next beam along the free edge. The curvature rate requirements are similar to those in (6.26) and (6.27) for 6 fields but the velocity u is, in general, not zero along the boundary.
VIII. Optimal Flexure Fields of Constrained Geometry
Sections IV-VII dealt with grillages and fiber-reinforced plates in which both the direction and cross-sectional area of the fibers are variable quantities. While the solutions obtained are of considerable theoretical interest, their geometrical complexity renders them somewhat uneconomical in actual design practice unless a large number of identical units are constructed or the total cost of a single structure is unusually high, as in the case
300
George I . N . Rozvany and Robin D . Hill
of aerospace projects or long-span terrestrial structures. In view of the foregoing, several geometrical constraints have been considered with a view to finding simple yet economical structural layouts for plane flexural systems. An early development (Rozvany and Charrett, 1971) was a study of optimal fiber-reinforced plates in which the specific reinforcement volume is continuously variable but the fibers are required to be straight and their direction is fixed. On the basis of (2.25) and (2.26), close analytical upper and lower bounds on the minimum fiber volume were obtained (Charrett and Rozvany, 1972b) and it was concluded that a simple upper bound approach called the " neutral line method " yields reasonably economical solutions. Another investigation (Rozvany et al., 1975b) considered the plastic design of prismatic grillages with beams in prescribed constant directions and the elastic design of prestressed plates having straight tendons terminated at the edges only. For these problems, rigorous analytical optimal solutions were tabulated for rectangular domains with various boundary conditions. A further research project (Rozvany et al., 1975a) looked into the optimization of fiber-reinforced plates having fibers in prescribed directions such that only a finite number of steps in the variation of the cross-sectional area of the fibers are permitted in the direction of the fibers and the reinforcing fibers have a constant length but variable spacing over a finite number of rectangular subdomains having unspecified boundaries. These constraints ensure a practical layout that is quite economical with regard to fiber volume. The last two topics were based on the general theories described in Sections II,E and I. One of the most significant recent developments has been proposed by Prager and concerns partially discretized grillages in which the direction of the beams is prescribed and only a finite number of main beams are used. The location and the number of main beams is to be optimized. The basic problem of partially discretized grillages will first be illustrated with a simple example. Example. Grillage Having a Discrete Main Beam and a Dense Set of Secondary Beams
The grillage shown in Fig. 81a is simply supported on three sides and free along the fourth side. Its beams in x direction have an infinitesimal spacing and the main beam in y direction is in an unspecified location. The grillage is subject to two line loads of intensity P in the positions indicated. All beams have a continuously varying cross section with specific cost functions k I M I and kl I M I for x and y beams, respectively, and the cost of connections
Optimal Load Transmission by Flexure
I:
30 1
a
3" (b)
K= k
Za--x-b SECTION C-D
(d) FIG.81. Partially discretued grillage.
along the beam AB is c I V I, where V is the shear force per unit width in the x beams, M the bending moment, and k and c constants. The moment diagram for the x beams is shown in Fig. 81b, where 6 = x(2a - .)/(a
+x)
(8.1) from statical considerations. The curvature rate condition for the beams is furnished by (2.13),the steps c in the velocity u of x beams (Fig. 81d) are given by (2.62),and the velocity A must be the same for both secondary and main beams (Fig. 8lc,d). Since in this problem the location of the support E is the same for all secondary beams and depends on the position of the main
George I . N . Rozvany 'and Robin D . Hill
302
beam AB, (2.72) must be integrated along the main beam for finding the optimal location of the latter:
which yields k[(2a - x
- b)'/2 - (2a - x - b/2)b] - kl a2/3 - c = 0,
(8.3)
where b is furnished by (8.1). For prismatic main beams, the velocity field satisfying (2.51)-(2.53) is shown in Fig. 83 and then the penultimate term in (8.3) changes to -kl a2/2. Optimal values of x are shown in Fig. 82 in which the line for kl = 0 is, naturally, valid for both prismatic and nonprismatic main beams. The problem of partially discretized grillages was introduced briefly elsewhere (Rozvany, 1975a, Part 11) and is discussed at depth in another paper (Prager and Rozvany, 1976) in which the optimal location and number of
I
I
I
0.5
I
I .o
I
1.5
I
1.66 c/ Ita'
2 .o
FIG.82. Optimal values of x in Fig. 81. Main beam: --, prismatic; -, nonprismatic.
Optimal Load Transmission by Flexure a
303
a ~
a2
h FIG.83. Velocity field for a prismatic main beam.
beams is given for square grillages having main beams parallel to the sides and also for a partially discretized layout similar to that given in Fig. 46a. In the study quoted, a finite number of beams in the optimal solutions is enforced by assigning the following specific cost function to the main beams: = alMl
+ b,
(8.4)
where a and b are constants. ACKNOWLEDGMENTS The authors are most grateful to Professor W. Prager for guidance and useful suggestions. Whereas both authors derived independently most of the significant findings of Sections IV-VI, the material in the other sections was contributed largely by the first author. The authors are indebted to Mr. D. Devenish (drafting), Mrs. J. Helm (typing), Mr. D. Holmes (photography), and Miss J. Zwolinska (computational and bibliographic work). LIST OF SYMBOLS? Basic variables generalized velocity (displacement) generalized load generalized strain rate (strain) generalized stress specific internal power (work) specific external power (work) spatial coordinate domain of a structure
P=
P= 9=
Symbols used in general optimality conditions
D, (e = 1, . ..*,
W)
i,(e = 1, ..., w ) P, ( h = 1, ..., t ) S, ( I = 1, ,.., r) +(
1
segments in partially prescribed cost distribution shape functions alternate loads stress regimes specific cost function
t For kinematic variables, the notation is given in the context of plastic design and the corresponding terms for elastic design are shown in parentheses.
George I . N . Romany and Robin D . Hill
304 (k = 1, ..., u )
41
cost components cost function for unspecified generalized reactions or loads cost function for joints Beams velocity (deflection) curvature rate (curvature) shear strain rate (shear strain) relative angular velocity (slope discontinuity, relative rotation) absolute angular velocity (slope) bending moment shear force generalized reaction or unspecified load force component of R couple component of R Grillages velocity (deflection) principal curvature rates (curvatures) circumferential and radial curvature rates (curvatures) rates of curvature and twist (curvatures and twist) in Cartesian coordinates principal moments circumferential and radial moments bending and twisting moments in Cartesian coordinates
REFERENCES BARNETT, R. L. (1961). Minimum weight design of beams for deflection. J . Eng. Mech. Diu., Proc. Amer. SOC.Civil Eng. 87, 75-109. CHARRETT, D. E., and ROZVANY, G. I. N. (1972a). Extensions of the Prager-Shield theory of optimal plastic design. I n t . J. Non-Linear Mech. 7 , 51-64. CHARRETT, D. E., and ROZVANY, G. I. N. (1972b). On minimal reinforcement in concrete slabs. Arch. Mech. 24, 89-104. CHERN,J. M.(1971). Optimal structural design for given deflection in presence of body forces. Int. J. Solids Struct. 7 , 373-382. CHERN,J. M., and PRAGER,W. (1970). Optimal design of beams for prescribed compliance under alternative loads. J. Optimir. Theory Appl. 5, 424431. CLYDE,D. H. (1967). Discussion of the paper by G. I. N. Romany (1966a). J. Amer. Concr. Inst. 64, 1551-1553. DAFALIAS, Y . F., and DUPUIS,G. (1972). Minimum-weight design of continuous beams under displacement and stress constraints. J . Optimiz. Theory Appl. 9, 137-154. D. C., and SHIELD,R. T. (1956). Design for minimum weight. Proc. Int. Congr. Appl. DRUCKER, Mech., 9th, Brussels, 5, 212-222.
Optimal Load Transmission by Flexure
305
DRUCKER, D. C, and SHIELD,R. T. (1957). Bounds on minimum weight design. Quart. Appl. Math. 15, 269-281. FOULKES, J. (1954). The minimum weight design ofstructural frames. Proc. Roy. SOC, Ser. A 223, 482-494. HAUG,E. J., and KIRMSERP. G. (1967). Minimum weight design of beams with inequality constraints on stress and deflection. J. Appl. Mech. 34, 999-1004. HEMP,W. S.(1973). “Optimum Structures.” Oxford Univ. Press (Clarendon), London and New York. HEYMAN, J. (1959). On the absolute minimum weight design of framed structures. Quart. J. Mech. Appl. Math. 12, 314-324. HODGE,P. G. (1959). “Plastic Analysis of Structures.” McGraw-Hill, New York. S.(1965). Economic design by the ultimate load method. Concr. Constr. Eng. LX KALISZKY, 365-372, 424-433, 461-467. LAMBLIN, D. O., and SAVE,M.A. (1971). Minimum volume plastic design ofbeams for movable loads. Meccanica 6, 157-163. LOWE, P. G, and MELCHERS, R. E. (1972). On the theory of optimal constant thickness fibrereinforced plates. Int. J. Mech. Sci. 14, 311-324. LOWE, P. G., and MELCHERS, R. E. (1973a). On the theory of optimal constant thickness fibre-reinforced plates, 11. I n t . J. Mech. Sci. 15, 157-170. LOWE, P. G., and MELCHERS,R. E. (1973b). On the theory of optimal constant thickness fibre-reinforced plates, 111. Int. J. Mech. Sci. 15, 711-726. LOWE, P. G., and MELCHERS, R. E. (1974a). On the theory of optimal, edge beam supported, fibre-reinforced plates. Int. J. Mech. Sci. 16, 627-641. LOWE,P. G.,and MELCHERS, R. E. (1974b).Some geometry of constant curvature surfaces. Proc. Cambridge Phil. SOC. 76, 601-605. MARCAL, P. V., and PRAGER,W. (1964). A method ofoptimal plastic design. J. Mec. 3,509-530. MARTIN,J. B. (1971). The optimal design of beams and frames with compliance constraints. Int. J. Solids Struct. 7, 63-81. MASUR,E. F.(1970). Optimum stiffness and strength of elastic structures. J. Eng. Mech. Diu., Proc. Amer. SOC.Civil Eng. %, 621-640. MASUR,E. F. (1974). Optimal structural design for a discrete set of available structural members. J. Comp. Methods Appl. Mech. Eng. 3, 195-207. MASUR,E. F. (1975). Optimality in the presence of discreteness and discontinuity. Proc. ILITAM Symp. Optimir. Struct. Des., 1973, Warsaw pp. 441-453. Springer-Verlag, Berlin and New York. W. (1967). Minimum-weight design of beams for multiple loading. MAYEDA,R., and PRAOER, Int. J. Solids Struct. 3, 1001-1011. MELCHERS, R. E. (1971). Optimal fibre-reinforced plates with special reference to reinforced concrete. Ph.D. Thesis, Cambridge Univ., Cambridge, England. MELCHERS,R. E. (1973). Optimal design of variable thickness reinforced plates. Civil Eng., Trans. Inst. Eng. Aust. IS, 99-102. MELCHERS, R. E. (1975a). Optimally reinforced axisymmetric plates. J. Eng. Mech. Diu., Proc. Amer. SOC.Civil Eng. 101, 143-149. R. E. (1975b). Optimal design of reinforced beams under alternate loading. Proc. MELCHERS, Australas. Con5 Mech. Struct. Muter., 5th Melbourne, pp. 393-404. MICHELL,A. G. M.(1904). The limits of economy of material in frame structures. Phil. Mag. [6] 8, 589-597. MORLEY,C . T. (1966). The minimum reinforcement of concrete slabs. Int. J. Mech. Sci. 8, 305-320. MORLEY, C. T. (1972). Discussion of the paper by Lowe and Melchers (1972). Int. J. Mech. Sci. 14, 903-905.
306
George I . N . Romany and Robin D . Hill
MROZ, 2. (1958). On the design of non-homogeneous technically orthotropic plates. Proc. I U T A M Symp., Warsaw pp. 191-202. Pergamon, Oxford. MROZ,2. (1967). On the optimum design of reinforced slabs. Acta Mech. 3, 34-55. MROZ,2. (1972). Multi-parameter optimal design of plates and shells. J. Struct. Mech. 1, 371-393. MROZ,Z., and ROZVANY, G. I. N. (1975). Optimal design of structures with variable support conditions. J. Optimiz. Theory Appl. 13, 85-101. F. G. (1970). On optimal design of reinforced annular slabs. Arch. Inz. MROZ,2, and SHAMIEV, Ladowej 16,575-584. NAGTEGAAL, J. C., and PRAGER, W. (1973). Optimal layout of a truss for alternative loads. Int. J . Mech. Sci. 15,583-592. PARKES, E. W. (1975). Joints in optimal framework. Int. J. Solids Srruct. 11, 1017-1022. W. (1957). On ideal locking materials. Trans. Soc. Rheol. 1, 169-175. PRAGER, PRAGER, W. (1967). Optimum plastic design of a portal frame for alternative loads. J. Appl. Mech. 34, 772-173. PRAGER, W . (1968). Optimal structural design for given stiffness in stationary creep. Z. Angew. Math. Phys. 19,252-256. PRAGER,W . (1970). Optimal thermoelastic design for given deflection. Int. J. Mech. Sci. 12, 705-709. PRAGER, W . (1971a). Foulkes mechanism in optimal plastic design for alternative loads. Int. J. Mech. Sci. 13,971-973. PRAGER, W. (1971b). Optimal design ofstatically determinate beams for given deflection. Int. J. Mech. Sci. 13,893-895. PRAGER, W. (1973). Methods of structural optimization. Int. Symp. Comp. Methods Appl. Sci. Eng., Versailles. Springer-Verlag, Berlin and New York. PRAGER, W . (1974a). “Introduction to Structural Optimization,” Courses and Lect. No. 212. Int. Cent. Mech. Sci., Udine. Springer-Verlag, Berlin and New York. PRAGER, W. (1974b). A note on discretized Michell structures. Comp. Methods Appl. Mech. Eng. 3, 349-355. PRAGER, W . (1975). Transmission optimale des charges par flexion. Ann. Inst. Tech. Batiment Trav. Pub. Suppl. No. 336. PRAGER, W., and ROZVANY,G. I. N. (1975). Plastic design of beams: optimal location of supports and steps in the yield moment. Int. J. Mech. Sci. 17,627-631. W., and ROZVANY, G. I. N. (1976). Optimal design of partially discretized grillages. PRAGER, J. Mech. Phys. Solids (to be published in Vol. 24). PRAGER, W., and SHIELD,R. T.(1967).A general theory of optimal plastic design. J. Appl. Mech. 34, 184-186. PRAGER,W, and TAYLOR, J. E. (1968). Problems of optimal structural design. J. Appl. Mech. 35, 102-106. REITMAN,M . I. (1967). Optimal design for three-dimensional reinforced concrete structures. (In Russ.). Vychisl. Organ. Tekh. Stroit. Proektir., Moscow 11, 40-44. ROZVANY, G. I. N. (1966a). A rational approach to plate design. J. Amer. Concr. Inst. 63, 1077-1094. ROZVANY,G.I. N. (1966b). Analysis versus synthesis in structural engineering. Civil Eng., Trans. Inst. Eng. Aust. 8, 158-166. ROZVANY,G. I. N. (1968). Optimal design of axisymmetric slabs. Civil Eng., Trans. Eng. Aust. lo, 111-118. ROZVANY,G . I. N. (1972). Grillages of maximum strength and maximum stiffness. Int. J. Mech. S C ~14, . 651-666. ROZVANY, G. I. N. (1973a). Optimum design for bending and shear. J. Eng. Mech. Div., Proc. Amer. Soc. Civil Eng. 99, 1107-1109.
Optimal Load Transmission by Flexure
307
ROZVANY, G. I. N. (1973b). Non-convex structural optimization problems. J. Eng. Mech. Dio., Proc. Amer. SOC.Civil Eng. 99, 243-246. ROZVANY, G . I. N. (1973~).Optimal plastic design for partially preassigned strength distribution. J. Optimiz. Theory Appl. 11, 421-436. ROZVANY,G. I. N. (1973d). Discussion of the paper by Lowe and Melchers (1972). Int. J. Mech. Sci. 15, 345-347. ROZVANY,G . I. N. (1973e). Optimal force transmission by flexure, clamped boundaries. J. Struct. Mech. 2, 57-82. ROZVANY, G. I. N. (1973f). Optimal topography of force transmission by flexure. J. Appl. Mech. 40,983-987. ROZVANY, G. I. N. (1973g). Optimal force transmission by flexure-the present state of knowledge. Proc. IUTAM Symp. Oprimiz. Struct. Des., Warsaw pp. 350-371. Springer-Verlag, Berlin and New York. ROZVANY,G. 1. N. (197311). Practical aspects of the optimization ofreinforced concrete structures. Con$ Concr. Res. Develop., Proc., Sydney pp. 7683. G. I. N. (1974a). Optimization of unspecified general forces in structural design. J . ROZVANY, Appl. Mech. 41, 1143-1145. ROZVANY, G. I. N. (1974b). Optimal plastic design with discontinuous cost functions. J . Appl. Mech. 41, 309-310. G. I. N. (1974~).Absolute optima in plastic design for preassigned shape. J. Appl. ROZVANY, Mech. 41, 813-814. G. I. N. (1974d). Closure on paper by Rozvany and Adidam (1972~).J . Eng. Mech. ROZVANY, Din, Proc. Amer. SOC.Cioil Eng. 100, 814-817. ROZVANY, G. I. N. (1974). Basic geometrical properties of optimal flexural force transmission fields. J . Struct. Mech. 2, 259-264. ROZVANY,G. I. N. (1974f). Discussion of the paper by Lowe and Melchers (1973a). Int. J. Mech. Sci. 16, 259-266. G. I. N. (1974g). Optimal flexure fields for corners. J . Eng. Mech. Din, Proc. Amer. ROZVANY, SOC.Civil Eng. 100, 828-831. ROZVANY,G. I. N. (1975a). Analytical treatment of some extended problems in structural optimization, parts I and 11. J . Struct. Mech. 3, 349-375, 377-392. G. I. N. (1975b). A unified theory of optimal moment fields. J . Struct. Mech. 3, ROZVANY, 179-195. . ROZVANY,G . I. N., and ADIDAM, S . R. (1971). On circular footing slabs. Build. Sci. 6, 41-44. ROZVANY, G . 1. N., and ADIDAM,S. R. (1972a). Recent advances in optimal plastic design. Proc. Int. Con$ Found. Plasticity, Warsaw pp. 201-218. Noordhoff, Groningen. ROZVANY, G. I. N., and ADIDAM, S. R. (1972b). Absolute minimum volume of reinforcement of slabs. J. Struct. Div., Proc. Amer. SOC.Civil Eng. 98, 1217-1222. ROZVANY, G. I. N., and ADIDAM,S. R. (1972~).Rectangular grillages of least weight. J. Eng. Mech. Dio., Proc. Amer. SOC. Civil Eng. 98, 1337-1352. ROZVANY, G. I. N., and ADIDAM,S . R. (1972d). Dual formulation of variational problems in optimal design. J . Eng. Ind. 94, 409-418. G. I. N., and CHARRETT, D. E. (1971). Slabs with variable straight reinforcement. J . ROZVANY, Struct. Dio., Proc. Amer. SOC.C h i [ Eng. 97, 1521-1530. ROZVANY, G. I. N., and COHN,M. Z. (1970). Lower bound optimal design of concrete structures. J. Eng. Mech. Dio., Proc. Amer. SOC.Cioil Eng. %, 1013-1030. G. I. N, and HILL,R. (1973). General theory of optimal topography of moment ROZVANY, fields, parts I and 11. Monash Univ. Civil Eng. Res. Rep. No. 6/1973. ROZVANY, G. I. N., and MELCHERS, R. E. (1970). Plastic design of axisymmetric slabs. Indian Concr. J. 44,201-206.
308
George I . N . Rozvany and Robin D . Hill
ROZVANY, G. I. N., and MROZ,Z. (1975). Optimal design taking cost ofjoints into account. J. Eng. Mech. Div., Proc. Amer. SOC.Cioil Eng. 101, 917-921. C. (1973).Grillages of least weight-simply ROZVANY, G. I. N., HILL,R, and GANGADHARAIAH, supported boundaries. Int. J. Mech. Sci. 15, 665-677. ROZVANY, G. I. N., GANGADHARAIAH, C., and HILL,R. (1975a). Optimal design of rectangular fibre-reinforced plates with piecewise constant reinforcement. Arch. Inr. Ludowej 21, 3- 14. ROZVANY, G. I. N., GANGADHARAIAH, C., and HILL, R. (1975b). Optimal slabs and grillages of constrained geometry. J. Eng. Mech. Din, Proc. Amer. SOC.Civil Eng. 101, 755-770. SACCHI,G., and SAVE,M. (1969). Le probleme du poids minimum, d‘armature des plaques en Wton arme. AIPC Extrait du volume 29“ a’es mimoires, Zurich, Switzerland. SAVE,M. A. (1972). A unified formulation of the theory of optimal plastic design with convex cost function. J. Struct. Mech. 1, 267-276. W. (1963). Minimum weight design of beams subjected to fixed and SAVE,M. A., and PRAGER, moving loads. J. Mech. Phys. Solids 20, 255-267. SAVE,M. A., and SHIELD,R. T. (1966). Minimum weight design of sandwich shells subjected to fixed and moving loads. Proc. Int. Congr. Appl. Mech., 11th Munich, 1964 pp. 341-349. W. (1968). Minimum-weight design with piecewise constant specific SHEU,C. Y.,and PRAGER, stiffness. J. Optimiz. Theory Appl. 2, 179-186. SHEU,C. Y.,and PRAGER,W. (1969). Optimal plastic design of circular and annular sandwich plates with piecewise constant cross-section. J. Mech. Phys. Solids 17, 11-16. SHIELD,R. T.(1960). Plate design for minimum weight. Quart. Appl. Math. 18, 131-144. SHIELD,R. T., and PRAGER, W. (1970). Optimal structural design for given deflection. J. Appl. Math. Phys. 21, 513-523. THAKKAR, M. C., and SRIDHAR RAO,J. K. (1970). Minimum reinforcement in clamped square slabs. J. Struct. Div., Proc. Amer. SOC.Civil Eng. %, 751-756. WATSON, H. L. (1973). Optimal design of space frames. Master’s Thesis, Monash Univ, Clayton, Victoria, Australia. WOOD, R. H. (1961). “Plastic and Elastic Design of Slabs and Plates.” Thames & Hudson, London.
The Role of Experiment in the Development of Solid Mechanics-Some Examples H.KOLSKY Division of Applied Mathematics Brown University Providence, Rhode Island
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Tensile Shock Waves in Rubber . . . . . . . . . . . . . . . . . . . V. Plastic Wave Propagation . . . . . . . . . . . . . . . . . . . . . VI. Experimental Studies in Dynamic Fracture . . . . . . . . . . . . . . VII. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . : . . . . . . . . . . . . . . . . . . I. Introduction
11. Viscoelastic Behavior . . . . . . . . . . . . . . . 111. Experimental Determination of Viscoelastic Response
309 316 322 345 349 354 364 365
I. Introduction This paper attempts to assess the role of experimental work in the development of the subject of solid mechanics. The author is inevitably influenced to a very considerable extent by his own experience as an experimenter, and most of the examples of recent work that are discussed in detail are concerned with experiments carried out by the author and his co-workers or by colleagues working on problems in closely associated fields. An excellent treatise, “ Experimental Foundations of Solid Mechanics,” which is historical in perspective and comprehensive in content, has recently been published by James Bell (1973).It is Volume VIa/l of the “ Encylopedia of Physics” and is over 800 pages in length. The scope of Bell’s book is obviously very much broader than anything that is being attempted here, 309
3 10
H . Kolsky
and the reader who wishes to pursue the detailed development of solid mechanics as an interaction between theory and experiment is strongly recommended to consult Bell’s authoritative and monumental work. The development of the experimental aspects of the subject has to a large extent been concerned with the measurement of the stress-strain response of real solids under various conditions of loading, or in the fashionable jargon, the determination of the constitutive relation of the solid. The oldest and simplest constitutive relation for solids is the threedimensional form of Hooke’s law, which for a general aeolotropic solid involves 21 distinct elastic constants but for an isotropic solid involves only two. Even for an isotropic elastic solid, however, the solution of the differential equations of equilibrium for specified boundary conditions is a matter of very considerable difficulty, and the many volumes devoted to the theory of elasticity give ample evidence of the difficulty of the subject and of the pathetically few problems for which solutions in closed form can be obtained. Only in physical situations with a very high degree of symmetry is it feasible even to attempt to obtain an analytic solution. The efforts devoted to the experimental and theoretical study of the photoelastic technique are a direct result of the enormous complexity of the analytical treatments when applied to real engineering structures, even where it may be assumed that Hooke’s law is a reasonable approximation to the mechanical response of the material of the structure. Experimental work in solid mechanics plays one of two distinct roles. The first is where the response of a material is assumed to be known (so that it is possible to write the constitutive relation between the stress tensor and the strain tensor) and where the boundary and/or initial conditions are well defined. The investigator is then faced with a perfectly well-posed mathematical problem that is just too difficult for him to solve by analytical means. Under these conditions the investigator may be forced to resort to experimental methods in order to solve his problem. He can do this either by attaching a large number of electrical strain gages to the structure itself (or to a model of it) and hence find the strain distribution in the structure, from which the stress distribution can be inferred. Alternatively, if the deformations are elastic he may make a model of the structure in a transparent photoelastic material, and by examining the fringe patterns in polarized light get information about the stress distribution in the model and hence in the structure itself. In these types of experimental investigations the experimental setup is acting as an analog computer. The equations of equilibrium for a given set of boundary conditions are being solved, but little new information about the nature of the response of the material to deforming forces is obtained. In this type of investigation the specimen geometry is generally complicated, and it is difficult to infer from the measurements what effect
Experiments in Solid Mechanics
311
deviations from the assumed stress-strain relation are going to have. Thus from a physical point of view this type of experimentation is not particularly valuable although, of course, it helps to solve the practical problems with which the mechanical engineer and the civil engineer are continually being confronted. The other type of experimental research in solid mechanics is primarily concerned with finding out how real materials behave under applied stresses, that is, what form the deviations from Hooke’s law take under the particular conditions of loading used in the experiment. In experiments of this type it is desirable to keep the geometry of the sample as simple as possible and to apply as simple a stress field as is consistent with the determination of the constitutive relation of the material under investigation. There are three types of deviation from Hooke’s law that are observed in real solids. The first is associated with the fact that a linear relation between the stress tensor and the strain tensor is only meaningful so long as the strain components have values that are small compared with unity. Thus the influence of the relative rotations of different parts of the structure on the stresses applied to them does not appear in the statement of Hooke’s law, and so long as displacement gradients are sufficiently small for their squares and higher powers to be neglected this type of formulation is self-consistent. When, however, we have situations where the strain components have magnitudes comparable with unity (e.g., in rubberlike elastic deformations), Hooke’s law clearly cannot be applied. This second type of deviation is geometrical in character and arises only when the strains are sufficiently large for the geometry of the structure to have been changed by the deformation. For many materials, however, deviations from linearity occur at quite small strains and arise not as a result of geometrical considerations but because of microscopic physical changes in the solid. Thus for most crystalline solids there is an elastic range, that is, a range of strains for which it is perfectly reasonable to use Hooke’s law, and then at some critical strain the material yields and further deformation results in deviations from linearity and the appearance of irrecoverable permanent strain. This type of deviation from linearity is the basis of the theory of plasticity of metals and of the various simplified models of plastic behavior that have been used to solve specific engineering problems. The third type of deviation from Hooke’s law is concerned with the effect of time on the relation between the stress components and the strain components of a physical structure. In ordinary engineering practice the dependence of the stress-strain relation on time shows itself in two ways. On the one hand, if a stress is maintained for a long time it is found that the deformation increases continuously with time; this phenomenon is known as creep. If, alternatively, a specimen of a material is deformed and
3 12
H.Kolsky
the magnitude of the deformation is kept fixed, the stress is found to decrease slowly with time; this phenomenon is known as stress relaxation. Both phenomena are aspects of what is known as viscoelastic behavior. The simplest type of viscoelastic behavior is called linear viscoelastic behavior and is described by Boltzmann’s principle of linear superposition. This principle states that the stress in a solid at any time t , depends not only on the strain c(tl) to which the solid is subjected at t, but on the whole strain history of the solid up to time t,. Furthermore, this stress can be expressed as a convolution integral involving the stress relaxation function and the strain rate for all values of time t from t = - 00 to t = t , . This type of response can be considered the other way around, where the strain ~ ( t ,is) treated as a function of the stress history of the sample for times up to t , . We then express the strain as a convolution integral involving the strain-relaxation function and the complete “rate of stress” history from t = -a3 to t = t,. We have here discussed viscoelastic behavior in terms of a single stress component and a single strain component but the generalization to three dimensions is comparatively straightforward, so that viscoelastic relations between the components of the stress tensor and the history of the separate strain components can be expressed as convolution integrals, except that in this case two distinct stress relaxation functions (e.g., one for dilatation and one for shear) are required. An alternative approach to linear viscoelastic response is in terms of linear differential equations involving the stress and the strain tensors and the derivatives of their components with respect to time. This type of treatment is generally written in terms of linear differential operators P and Q, where each of these operators is a polynomial in the operator D = d/dt. Here again, however, the three-dimensional form involves two pairs of operators, P and Q for shear deformation and P‘ and Q for volume changes. All real solids exhibit deviations from Hooke’s law of all three types when they are deformed, but fortunately in many practical situations it is possible to ignore at least two of the deviations. We have, for example: (i) the deformation of some rubberlike solids where the magnitudes of the strains are comparable with and often greater than unity, but where viscoelastic effects and irrecoverable flow can be neglected ; (ii) the deformation of metal structures where plastic yield is important but where the strain components are nevertheless small compared with unity and where time effects can be neglected; (iii) linear viscoelastic deformation of engineering structures of polymer
Experiments in Solid Mechanics
313
solids, where Boltzmann’s principle of linear superposition gives a close representation of the observed mechanical response. A large part of the contribution of experimental work to the interpretation of the problems of solid mechanics is concerned with measurements of the stress-strain behavior of specimens of various solids, which are carried out with the object of assessing the nature of the deviations from ideal behavior, both with respect to nonlinearity and with respect to time dependence. These are sometimes carried out over a range of ambient temperatures. When the time dependence of the stress-strain relation can be ignored, the experimental techniques employed are fairly straightforward at least insofar as simple extension and shear response are concerned. With rod-shaped specimens the extensional response can be determined in various types of extensional testing machines. These give the extension as a function of the applied longitudinal stretching force over a range of extensional strains. The easiest method of determining the shear response of a material is by measuring the twist of specimens when they are deformed torsionally. In order to ensure that the shear deformation is effectively uniform over the whole of the specimen, it is necessary for its shape to be in the form of a thin-walled hollow cylinder. When it is necessary to study the response of solids to volume changes, more complicated apparatus is required and the presence of voids in the specimen itself, or of air bubbles in a fluid used to apply the pressure, may lead to erroneous measurements. In the presence of nonlinearities it is often required to study the interaction between the behavior of the different strain components and here we are faced with the problem that even where measurements are carried out under conditions of ordinary atmospheric pressure, it is often a matter of some difficulty to design a testing machine that will accurately determine the response under conditions of combined tension and torsion and allow each to be varied separately. When we come to the determination of the time dependence of the stressstrain relation a mass of new difficulties arises. For example, the simplest type of timedependent measurement is that of extensional creep, i.e., the dependence of the magnitude of the extensional strain as a function of the time for which a longitudinal stress has been applied to a rod specimen. Ideally to do this experiment we need to apply a stress that will be kept constant with time and observe how the length of the specimen increases with the length of the duration for which this constant stress has been applied. To begin with, if a load is applied suddenly to a specimen, oscillations are set up, and measurements made before these oscillations have effectively died down are difficult to interpret. Consequently, the direct
3 14
H . Kolsky
measurement of creep behavior for very short times is a matter of some difficulty even when measuring appara.tus is available for recording the extensional strains at very short times after the load has been applied. Further, if the strain is finite the cross section of the specimen is less than it was before the load was applied; consequently the true stress is greater than the engineering stress and as creep progresses the true stress continues to grow as a result of the decreasing cross section of the tensile specimen, Various mechanisms have been proposed to compensate for this effect. Perhaps the most ingenious is one proposed by Andrade (1914), in which the weight that produces the extensional load dips into a liquid of suitable density and is so shaped that as the specimen extends the increasing upthrust of the liquid on the weight reduces the extensional load to an extent that just compensates for the decreasing cross section, and keeps the value of the extensional stress constant. The complementary time-dependent measurement is that of stress relaxation. Here the object is to keep the extensional strain constant and then measure the stress as a function of time of loading. The problems here are once again associated with the determination of the short time response since mechanical vibrations are liable to cause errors, and although there is now no change in cross section with time, it is difficult to make stress measurements and at the same time keep the length of the specimen fixed. Thus, however rigid the stress-measuring apparatus may be, it will tend to allow the specimen to increase in length as the stress relaxes. With specimens of comparatively soft materials this effect is often negligibly small, but for more rigid materials either a servomechanism must be used to restore the specimen to the fixed length as stress relaxation proceeds, or more simply the specimen is stretched with weights and maintained at constant length manually by removing weights as the stress relaxes. When we are dealing with materials whose mechanical response is time dependent but which are linear (in the sense that relations between the stress components and the strain components can be expressed as linear differential equations with respect to time), mechanical oscillations furnish a very convenient method of determining the short-term response. Thus there is generally a range of frequencies where useful determinations can be made by oscillating a specimen at a fixed frequency and observing the stress amplitude required to produce a given strain amplitude. Two quantities are determined in such measurements: the ratio of the stress amplitude to the strain amplitude, and the phase difference between the oscillating stress and the oscillating strain. For an elastic material the ratio of these two amplitudes is independent of the frequency of oscillation and gives the elastic modulus of the solid for the type of deformation that the sample is undergoing (e.g., extensional,
Experiments in Solid Mechanics
315
torsional, flexural). For elastic samples there is also no phase difference between the oscillating strain and the oscillating stress. For linear viscoelastic materials, however, the ratio of stress amplitude to strain amplitude increases with increasing frequency of oscillation and the oscillations in strain lag behind the oscillations in stress. The frequency dependence of the ratio of the amplitudes of the stress and the strain, and the phase lag of strain behind stress define the viscoelastic response of the solid. The magnitude of the phase lag is often taken as a measure of the internalpiction of the solid and, particularly in metals but also in polymers, this can often be related to specific microscopic processes that are taking place as a result of the mechanical oscillations. For materials that are not too highly dissipative the phase lag can be simply related to a number of other measures of internal friction such as the specific mechanical loss (this is the fraction of the mechanical energy that is dissipated in a mechanical cycle), the mechanical Q, the logarithmic decrement, the sharpness of resonance, and the attenuation of stress waves propagated through the material. We now return to listing the other experimental techniques that can be used for measuring the time-dependent mechanical response of materials. Whereas the direct observation of the oscillating strain and the oscillating stress on a specimen that is producing it is probably the best method of making measurements at very low frequencies ( < 1 Hz), at somewhat higher frequencies it is more convenient to study the free oscillations of a mechanical system, which consists of a large rigid inertia (large in the sense that its magnitude is very much greater than that of the specimen under investigation) connected to a viscoelastic specimen; the specimen acts as the elastic restoring force. From measurements of the period of such free oscillations and of their rate of decay, the viscoelastic constants of the specimen at the frequency of the oscillation can be determined. The same type of arrangement can be observed under conditions of forced oscillation and here the resonant frequency and the " half-breadth " of the resonant peak provide the necessary data. In the two types of measurement outlined above, the assumption is that all the inertia of the system is concentrated in the rigid auxiliary element and the inertia of the specimen itself can be ignored. This is equivalent to assuming that the wavelength of stress waves at the frequency of the oscillation is very large compared with the length ofthe specimen. Under these conditions the stress and strain in the specimen at any given instant are distributed evenly along its whole length. At frequencies for which this is no longer true, wave propagation in the specimen must be taken into account. The lowest natural frequency that can be observed with an unloaded rod specimen is where the wavelength is equal
3 16
H.Kolsky
to four times the length of the specimen. This frequency of vibration corresponds to free oscillations of the specimen when one of its ends is fixed. These oscillatiQns are generally called standing waves ; of course, higher harmonics can also be excited. For such standing-wave oscillations once again either the.period and the logarithmic decrement of free oscillations can be measured, or the resonance peak can be recorded by studying the response to a constant driving source that sweeps through the resonant frequency. At still higher frequencies the propagation of stress waves must be observed directly, and here the phase velocity of propagation and the attenuation coefficient are the parameters that must be measured for wave trains of different frequencies.
11. Viscoelastic Behavior Although the discussion of the time dependence of the stress-strain relation for real solids goes back at least to Coulomb (1784), it was not until Boltzmann (1876) formulated his principle of linear superposition that any consistent theory of time-dependent mechanical response was available. Boltzmann postulated that for all real solids the relation between the stress components and the strain components at any given time depended not only on their instantaneous values at that time but on the whole history of the deformation from the time that the material was first formed. Further, Boltzmann postulated that for many real materials the stress at any time resulting from a series of deformations that had taken place at earlier times was a simple sum of the stresses that would have been produced had the separate deformations occurred singly. To illustrate the principle mathematically, let us assume that a uniaxial strain el occurred at time tl and was maintained; then at time t this strain will produce a stress ol(t - t , ) . Similarly, if instead a strain e2 occurred at time t , and was maintained, then at time t the stress produced by it will be 0 2 ( t - t 2 ) . Boltzmann’s principle states that if the strain history consisted of a strain that occurred at tl and an additional strain E, that occurred at t,, then the stress a(t) at time t is given by a(t) = ol(t - t l ) 0 2 ( t - t 2 ) . This type of linear superposition can be extended to any number of strain increments. Now the function that defines the way the stress decreases with time for unit strain is called the stress relaxation function, and we shall call this function for simple extension E ( t ) ; this means that if an extensional strain E is applied at time t = 0 and maintained, then the stress at time t is given by eE(t).
+
Experiments in Solid Mechanics
3 17
If the strain history consists of a series of steps E, each applied at time t = t,, the stress o(t) observed at time t will, according to the principle of
linear superposition, be given by n
o(t)= C &,E(t - t,). 1
Now any continuous strain history ~ ( tcan ) be expressed as a sum of infinitesimal strain increments AE" taking place at times t, so that ~ ( tz) C; H ( t - t,) AE, [where H ( t ) is the unit step function at t]. Boltzmann's principle states that the stress history corresponding to the right-hand side of this relation is n
2 E(t - t,) AE, .
~ ( t=)
1
In the limit as AE, -+ 0 this can be expressed as an integral, and we have tn=t
~ ( t=) tn=
- a0
E ( t - t,) d e n ,
or we can write the equivalent relation E(t - r)(d&/ds)ds.
o(t) = r=-m
This is one form of Boltzmann's principle. We may instead treat the problem as one where the specimen undergoes a given stress history a(t) and consider an expression for the strain. If a unit stress is applied at time t = 0, the strain at time t is given by the creep function, and for extensional strains we denote this by Jft). An exactly similar application of the principle of superposition gives the total strain ~ ( tat) time t in terms of a convolution integral in which the kernel is the creep function and the total strain is expressed as ~ ( t=) J r = '
J ( t - t)(do/dr) d t .
r=-m
Similar superposition expressions can be written for shear deformation and for volume changes, and relations that parallel the three-dimensional expression of Hooke's law can be formulated for general three-dimensional linear viscoelastic deformation, except that we must now have two kernels: one for dilatational strains (corresponding to the bulk modulus in Hookean elasticity) and one for shear strains (corresponding to the rigidity modulus in elasticity).
H.Kolsky
318
This formulation of linear viscoelastic behavior, although more satisfactory in that the physical assumptions are logically more acceptable, is mathematically rather cumbersome, and other equivalent representations that are less physically appealing but mathematically rather more convenient have received much more attention in the literature. One of these is to represent the stress-strain relation in terms of differential operators; thus for a single stress component cr and the corresponding strain component E the stress-strain relation may be written
Pa = QE, where P and Q are linear differential operators that can be expressed as polynomials in D,where D is the operator d/dt. Thus
P = a.
+ u , D + a z D 2 + + a,D” + ...
and
Q = bo + b , D + bz D z +
+ b,D” + .-.,
where the a’s and b’s are constants. If P and Q are chosen to be polynomials of sufficiently high degree, this type of representation can be made to correspond as closely as one desires to the integral form used in Boltzmann’s treatment, which involves the stress relaxation or the creep function. Once again, the response in shear and the response in dilatation must, however, be treated separately and two pairs of differential operators are required for three-dimensional problems. However, the mathematical treatment of differential equations of high degree is no simple matter, and in most problems that have been treated analytically the degree of P and Q that are used is very rarely higher than a quadratic. As is shown later, this concession to mathematical tractability has generally resulted in a considerable loss in physical reality. Now the fact that for linear viscoelastic solids the relation between stress and strain can be expressed as a convolution integral of the rate of strain or as a linear differential equation with constant coefficients implies that if the strain is varying sinusoidally with respect to time at some angular frequency w, then the stress will oscillate with this same angular frequency, but need not necessarily be in phase with the stress. For real viscoelastic solids the strain lags behind the stress, so that if we have ~ ( t=) E~ cos wt, then a(t) = cro cos(wt + 6), where e0 and cro are the amplitudes of the oscillations. In general, both the “modulus” cro /co and the phase angle 6 will vary with by the symbol IE*l and for real the frequency w. We denote a o / ~ O viscoelastic solids I E* 1 increases with increasing w, while 6 can either increase or decrease and in general shows a number of flat peaks when plotted over a sufficiently wide frequency range.
Experiments in Solid Mechanics
3 19
A convenient way of treating this type of “dynamic modulus” is in terms of complex notation; thus if we express the strain E ( t ) as ~ ( t=) R[E, exp iwt],
we have a(t) = R{a,
exp(iwt + 6)).
Now we define a complex modulus E* = El + iE2 that relates the complex quantities on the right-hand sides of these two equations. Thus we have 6,
exp i(wt + 6) = &,(El + iE,) exp iwt,
so that El + iE, = (a0/~,)[cos6 + i sin 61. Thus E l = I E* I cos 6 and E, = I E* I sin 6. We therefore have tan 6 = E , / E , and
I E* 1 = (E: + E$)”2. Now it is fairly straightforward to determine the values of IE*I and 6 experimentally over wide ranges of frequency for real materials by the use of the series of techniques outlined in the introduction to this article. This will be discussed in detail later in this section. It is also fairly straightforward to derive analytic expressions for these quantities for various choices of the polynomials P and Q. A comparison between the experimentally determined values and those predicted theoretically makes it possible to assess the range of validity of the assumed constitutive relations. Before going into this, however, we will further discuss some other aspects of the theory of linear viscoelasticity. As was mentioned above, in practice the differential operator approach to describe the viscoelastic response of real solids involves using very simple expressions for the operators P and Q. Thus the very simplest approach is that a, and b, are both finite constants, and all the other a’s and b’s are zero. This gives simple elastic behavior where the elastic modulus is b, / a o .Perhaps the next simplest assumption is that P = a, and Q = b, D. This assumption is equivalent to simple Newtonian viscous flow with the coefficient of viscosity q equal to bl / a o . A slightly more complicated form is to take P = a, and Q = b, + b,D. This corresponds to what is called the Voigt or Kelvin solid and the constitutive relation of such a solid is often written o=E E
+ q’(ds/dt).
(2.1) Such a solid behaves like a mechanical model that consists of a spring of modulus E‘ joined in parallel with a “dashpot” of viscosity q‘ as shown in Fig. 1.
H . Kolsky
320
FIG.1. (a) Kelvin-Voigt and (b) Maxwell models.
Another simple type of viscoelastic solid is one where P = a. + a , D while Q = b , D. This is generally referred to as a Maxwell material and has the stress-strain relation (ofz)
+ (do/dt)= E defdt.
(2.2)
This material behaves like a mechanical model that consists of a spring of modulus E joined in series with a viscous element of viscosity q, where q = ET. T is the relaxation time of the material for reasons that will be described later. Such a model is also shown in Fig. 1. A slightly more complicated model is one where both P and Q are linear, i.e., P = a. + a, D and Q = bo + b, D.Thus the equation of this solid is of the form
a o o + a, dufdt = boe -k b , defdt.
(2.31
Such a material is called a standard linear solid. This solid can be represented by one of two equivalent mechanical models, namely, a Maxwell element with a spring across it or a Kelvin element with a spring in series. Both models are shown in Fig. 2. As will be shown later, all these models are very poor representations of the behavior of most real linear viscoelastic solids except over very restricted frequency ranges, and in order to provide an adequate description the polynomials P and Q need to be of quite high degree. An alternative approach to such more complicated response, however, is to describe the stress-strain response as equivalent either to a large number of Maxwell elements joined in parallel, or to a large number of Kelvin-Voigt elements joined in series.
Experiments in Solid Mechanics
32 1
(b)
(0)
FIG.2. Two models of standard linear solid.
For a Maxwell material a unit strain applied at time t = 0, i.e., E(t) = H(O),
results in a stress a(t), which is given by a ( t ) = E exp( - t/z).
(2.4)
But the stress at time t produced by unit stress applied at time t = 0 is what we have called the stress relaxation function E(t). Thus for a single Maxwell element, E(t) = E exp( -t/r),
(2.5)
where z is the time it takes for the stress to fall to lle of its initial value. For two Maxwell elements joined in parallel one of which has a modulus E and a relaxation time z while the other has a modulus E' and a relaxation time z', it is simple to show E(t) = E exp( - tlr)
+E
exp( - tlz'),
(2.6)
and for a continuous distribution of Maxwell elements we may write E(t) =
5
F ( z ) exp( - t / z ) d In z.
0
F ( z ) is called the relaxation time spectrum of the material and peaks in this spectrum often correspond to specific mechanical processes that take place
H. Kolsky
322
in the solid. Similarly a retardation time spectrum for the distribution of Kelvin-Voigt elements can be derived to describe the creep function of a linear viscoelastic solid, and sometimes it is this spectrum rather than the relaxation time spectrum that can be related to the underlying microscopic physical processes going on in the viscoelastic solid. If we consider the three simplest linear viscoelastic materials, namely, those corresponding to the Kelvin-Voigt, the Maxwell, and the standard linear solid, we find that for the Kelvin-Voigt material tan 6 = WT’, where w is the angular frequency of the oscillation and T‘ = q/E, while I E* 1 = E(1 0 2 T ’ 2 ) 1 / 2 . For the Maxwell solid,
+
tan 6 = ~ / o T , while
I E* I = EON(1 + w 2 ~ 2 )112.For a standard linear solid the algebraic expressions for tan 6 and I E* 1 are rather more complicated; however, if we put T = a, /ao in Eq. (2.3) ( T is in fact the relaxation time of the Maxwell element when we consider the standard linear solid to be represented by a spring connected across a Maxwell element), we find that at frequencies w for which OT << 1, the material behaves like a Kelvin-Voigt solid and tan 6 is proportional to o,whereas if COT >> 1, the material behaves like a Maxwell solid and tan 6 is inversely proportional to w . In covering the frequency spectrum, we also find that at low frequencies (COT << l), the value of I E* I is given by bo / a o , while at very high frequencies it is equal to b, /al.
111. Experimental Determination of Viscoelastic Response
In the introduction to this article we outlined various experimental techniques that have been used for the determination of the linear viscoelastic response of materials. At very low frequencies a direct method is to observe the oscillating strain as E = E~ cos wt and the corresponding oscillating stress that produces it, namely, o0 cos(wt + 6). The ratio of the stress amplitude to the strain amplitude gives the value of I E * ( at the frequency w, while the phase angle can be observed directly. An apparatus that enables measurements of this type to be made conveniently for the linear viscoelastic response in shear is shown in Fig. 3. This type of apparatus has been used by Lethersich (1950), Markovitz et al.
Experiments in Solid Mechanics
323
SYNCHRONOUS MOTOR and 12 SPEED
GEAR
BOX
' TORSION HEAD
WIRE OF KNOWN STIFFNESS
THERMOSTATED G L A S S WATER
ts FIG.3. Apparatus for measuring linear viscoelastic properties at low frequencies.
(1952), Benbow (1954), and others. It can be employed at frequencies that are sufficiently low for the inertia of the apparatus to be neglected. In this apparatus the specimen is in the form of a rod, the lower end of which is fixed while the other end is oscillated by an oscillating torsion head that is connected to it through a metal wire of known torsional stiffness. The head is driven by a synchronous motor through a 12-speed gear box and a harmonic transformer, which converts the rotation into an oscillatory motion.
H . Kolsky
324
The measurements were made by reflecting collimated light beams from two mirrors, one of which is attached to the oscillating torsion head while the other is attached to the top of the specimen. The reflected light beams were observed with conventional galvanometer lamp and scale systems. From the oscillations of the mirror attached to the top of the specimen, the magnitude and phase of the shear strain can be inferred if the dimensions of the rod specimen are known, while the amplitude and phase of the oscillating stress can be determined from the difference in deflections of the two mirrors since this gives the angular twist of the wire of known stiffness. With this type of apparatus measurements can be conveniently made in the frequency region 1 cycle per day to about 2 cycles per second. At frequencies very much higher than this, recording the oscillations becomes more difficult and inertia effects begin to invalidate the accuracy of the method. At these higher frequencies the methods of free oscillation and of resonance are the ones generally used to make viscoelastic determinations, and for free oscillations the viscoelastic parameters are determined from the period of the oscillations while the value tan 6 is given by the value of the logarithmic decrement A . Thus, for simplicity let us consider a mass M that is attached to a rod of length 1 and cross-sectional area A. If the mass is displaced a distance u, the longitudinal strain in the specimen, assuming that it is uniform, is ull. Thus if we assume that the specimen is oscillating at a frequency w we may write in complex notation E ( t ) = e0
exp iwt,
so that u ( t ) = uo
exp iwt = bo exp iwt,
where uo is the amplitude of the oscillation. We then have that the stress o(t) = (El + iE2)&(t), so that a(t) = (El + iE2)e0 exp iot, and the force acting on the mass M is Aa.Thus applying Newton’s second law to the mass M, we have A0 = - M d2U/dt2, or A(E1 + iEZ)eOexp iwt = Mw21eo exp Thus, o2= K(EI
+ iE2),
where K = A/Ml.
iot.
325
Experiments in Solid Mechanics
If E , is zero, the material is elastic and the angular frequency w is then [ E l K]”’. If, however, E , is finite w must be complex, i.e., w = w1
+ iw, .
This means that the displacement u is given by u = R[u,
exp i(wl
+ i o , ) t ] = uo exp - w , t
cos w1 t .
(34
Thus the displacement is damped exponentially with respect to time. Let the amplitude of the nth oscillation be u, and the amplitude of the (n + 1)th be u,+ Then the logarithmic decrement A’ is defined as ln(u,/u,+ 1) and from Eq. (3-1) it can be seen that
A’ = 2xw2/w1. Since u 2 / K = K[(wI - ma) + 2i01w,]/K = E ,
+ iE, ,
then El = (0: - w i ) / K ,
(3.2)
E , = ~u,u,/K. (3.3) For most real viscoelastic solids E 2 << E l and consequently w2 << w l . Now, tan 6 = E 2 / E 1 = 201w2/(o: - 0:)
= 2w2/w1,
so that
A’ = 2x02/w1 = x tan 6. Thus tan 6 = A/x.
(3.4)
The period of the oscillation T = 2z/m1, and once again since 0, << wl, we also if tan 6 << 1, have (wf - w : ) N 0:;
I E* I
N
El(l
+ tan’
8)”’
N
El,
so that w : = E 1 K = IE*IK.
Thus
I E* I
= w : / K = 4z2/KTZ= 471’M1/AT2.
(3.5)
Figure 4 shows an apparatus that has been used to measure the shear
326
H . Kolsky
I N E R T I A DISK
DETECTOR
-THIN
SUSPENSION
BASE
FIG.4. Torsional oscillation apparatus.
response of linearly viscoelastic samples both by the method of free vibrations that was outlined above, and by the resonance method, which will be discussed next. In order to produce purely shear deformations, the specimen is deformed in torsion. A heavy horizontal metal disk of moment of inertia I is supported by a thin vertical metal wire that is on the one hand strong enough to support the weight of the disk, and on the other has negligible torsional rigidity, so that it produces only a very small restoring couple when the disk is twisted. The lower face of the disk is attached to the top surface of the viscoelastic specimen, which is in the form of a cylinder of length 1 and radius a. The bottom face of the cylinder is attached to a heavy rigid base. The oscillations are set up by passing an alternating current through a coil that was attached
Experiments in Solid Mechanics
327
to a point on the circumference of the metal disk and moves in the field of a strong permanent magnet. By varying the frequency of the driving current through the coil the resonant frequency can be found, and when oscillations had been set up at this frequency the current was switched off so that the oscillation decayed. To observe the oscillations, a condenser microphone detector was employed. This type of detection is highly sensitive and does not load the oscillating system. The oscillations of the disk resulted in an oscillatory value of the capacity between the detector plate and the grounded plate of the capacitance detector, and this varying capacity was used to frequency modulate an rf oscillator. In order to obtain a record of the oscillations, the fm signal had to be demodulated, which was done by standard fm techniques with the use of a frequency discriminator. Now the sinusoidal oscillations of the system are given by the equation w 2 ~ exp o iot = K ( G 1
+ ~ G , ) exp E ~ iot,
where K' = $na4/ll and G 1 + iG, is the complex shear modulus at the frequency of the oscillation. The value of the phase lag 6 is given as before by the relation tan 6 2: nA', where A is the logarithmic decrement, while the value of 1 G* 1, the ratio between the stress amplitude and the strain amplitude, is given by
1 G* I
N
G 1 % w ; / K = 8nll/T2a4,
where T is the period of the oscillations. It should perhaps be pointed out here that since we are observing a damped oscillation, the use of a complex modulus G 1 + iG, at a single frequency w1 is not really valid. The Fourier spectrum of a damped sinusoidal functionis not a single line but has a finite spread. The bandwidth Aw, i.e., the frequency spread for half-peak amplitude in the spectrum, is in fact given by Aw/o
2:
A/n
for A'
1.
Consequently, since A is usually a small fraction, the bandwidth is small and this effect may be neglected. This type of measurement has been carried out by a number of workers, e.g., Benbow (1954). We will now consider the forced oscillations of a system such as that shown in Fig. 4. If the forcing function is a couple C(t), the equation of motion may be written I d28/dt2+ R ( t ) = C(t),
(34 where R ( t ) is the net restoring couple in the specimen at time t. In general, Eq. (3.6) is difficult to solve for a genera1forcing function. If,however, C ( t )is
H . Kolsky
328
an oscillating couple, namely, C ( t )= .Coexp ipt, the response of the system will consist of a transient phase that will decay, followed by a steady-state response where 8 will oscillate sinusoidally with the angular frequency p but not necessarily in phase with C(t),i.e.,
8(t) = 8, exp i(pt - a), where 8, is the amplitude of 8 and 6' is the phase lag of 8 behind C. Thus the equation of motion of the system for the sinusoidal oscillations, once the transient conditions have decayed, is given by
+
Co exp ipt = -Zp28, exp i(pt - 6') ( G ,
+ iG2)K'100exp i(pt - a),
where Z and K' have the same values as for the free oscillations, so that Co exp id' = [(G,K' - p')
+ iG,K']18, .
(3.7)
Equating real and imaginary parts in Eq. (3.7) we get Co cos 6' = Z[G1K' - p2]80
C, sin 6' = Z[G,K']8,
.
(3.8) (3.9)
From (3.8) and (3.9) we obtain the response equation
(O,),
= Ci/Z2{(G, K'
-p2),
+ (G2K'),}.
(3.10)
This is a resonanoe-response curve and the maximum amplitude occurs at an angular frequency p o , where
p i = G 1K', so that
1 G:J = G~ = p ; / K
= 2p31~na4.
The amplitude 8, will fall off at forcing frequencies away from po and the halfbreadth of the resonance curve is defined by the quantity ( N , - N , ) / N = A N / N , where N , and N , are the frequencies on the two sides of the resonant frequency N (= po/2n),at which 8, has half its maximum value. From Eq. (3.10) it can be shown that this half-breadth is proportional to tan 6; thus
AN/N
2~
f i tan 6.
Thus observations of the position and sharpness of the resonance peak enable one to determine the complex viscoelastic modulus at the resonant frequency. As was pointed out earlier, the complex modulus is a function of the frequency, so that strictly speaking the assumption of constant values for
Experiments in Solid Mechanics
329
G, and G, over the frequencies covered in the determination of the resonance curve is not justified. These quantities, however, vary only very slowly with frequency and the assumption that their values are constant in the range p , to p , introduces only very small errors in the determination of the phase angle 6. A more serious error is often introduced by the mechanical coupling between the driving coil and the permanent magnet; this effect results in a broadening of the resonance peak (Davies and James, 1934) and may be allowed for in making determinations of the half-breadth. One way of doing this is by reducing the strength of the permanent magnetic field and thus extrapolating to the situation where the mechanical coupling is vanishingly small. The methods of free oscillation and of resonance cover much the same frequency range, which is from a few hertz to several kilohertz. The method of free oscillations is most suitable when the damping is low, while the resonance method is most effective when high damping is present. In the treatment of both of these methods, it has been assumed that the inertia is lumped in a rigid auxiliary mass and that the inertia of the specimen itself can be neglected in comparison with that of the mass. When this is so, the frequency of the oscillation is such that the wavelength of stress waves of that frequency is very large compared with the length of the specimen. At frequencies for which this no longer applies, the analysis has to be treated in terms of wave propagation. This is discussed below for linear viscoelastic solids. In order to illustrate the nature of stress wave propagation in linear viscoelastic solids we will consider the simple problem of extensional wave propagation along a thin linearly viscoelastic rod. The rod is assumed to lie parallel to the x axis and we derive the equation of motion of a thin element of the rod between the sections at x and x + Ax. Let the longitudinal displacement of the section at x, assumed uniform over the cross section, be u and the longitudinal stress on the section, also assumed to be uniform, be u. Let the density of the rod be p and its cross-sectionalarea be A, so that the mass of the element is p A Ax. If Ax is sufficiently small the acceleration in the x direction will be d2u/at2and the net force in the direction of increasing x will be A(aa/ax)Ax, so that the equation of motion may be written A AX aa/ax = P A AX a2u/at2
or
aolax = a2u/at2.
Differentiating partially with respect to x we derive the partial differential equation aZg/ax2
=p
a2Ejat2,
(3.1 1)
where E = dujdx, the extensional strain. This equation is quite general and
H. Kolsky
330
the mechanical response of the material of the rod has so far not been considered. If the rod is linearly viscoelastic and if we assume that an infinite train of sinusoidal waves of frequency o is propagated along it, then as shown earlier we may write CT
= (El
+ iE&,
since each element of the bar is undergoing a longitudinal sinusoidal oscillating strain of frequency o.Making this substitution for Q in Eq. (3.11) we obtain
( E , + iE2) d2E/dt2 = p
a2&/at2.
(3.12)
Equation (3.12) is the one-dimensional wave equation where the velocity C is complex, i.e., a2&/at2= cza2u/ax2 =
(c,+ iCz)2a2E/axz,
where
C = C1
+ iC2 = ( ( E l + iE2)/p)1’2.
(3.13)
Thus for the propagation of a sinusoidal disturbance, we have E
=R
[ E exp ~ i(ot - Sx)],
(3.14)
where fi = w/C. Since B is complex, we may write
j? = o / c - ia. Thus (3.14) becomes E
= E,,
exp( - a x ) cos[o(t
- x/c)],
(3.15)
where ct is called the attenuation constant and c is the phase velocity of propagation of the train of longitudinal sinusoidal waves of frequency w. Now = w/C = w/(C1
+ iC2) = w(C1 - iC2)/I C 12.
(3.16)
However, j? = w/c - ia, so that equating real and imaginary parts we find c = ICIz/C,
and
a =oC,/JC~~.
(3.17)
From equations (3.13) and (3.17) we finally derive c =(
I E* I / P ) ” ~ sec 612
and
a = ( o / c ) tan 6/2,
(3.18)
where 6 is the phase angle by which the strain lags behind the stress.
331
Experiments in Solid Mechanics
When 6 << 1, as it is for many linear viscoelastic solids, the relations
1
c = ( E* / / p ) ’ I 2
and
a = w tan 6/2c
(3.19)
are sufficiently accurate for most practical purposes. It is possible experimentally to determine c and a directly by propagating a sinusoidal wave train along a long rod or filament of a viscoelastic solid and make observations of the phase and the amplitude of the oscillations at a number of points along the rod. Thus if the oscillation at the source (which we may take to be the origin x = 0) is uo cos wt, the oscillation at a distance x along the rod is given by u(x, t ) = uo exp( - a x ) cos(o(t
- x/c)).
(3.20)
Thus if we observe the oscillation at x’ to be u(x’, t) = u9 cos(wt - $),
we have $ = wx’/c
and
us = u,, exp( -ax’).
(3.21)
From observations at two or more stations we can readily find a and c, and hence with the aid of Eqs. (3.19) [or, if greater accuracy is required, (3.18)] I E* I and 6 can be determined for the frequency of the oscillation w . An apparatus for doing this, which has been used by Hillier and the author (Hillier and Kolsky, 1949), is illustrated in Fig. 5. In this apparatus the driving unit is an electromechanical transducer. At audio frequencies this can be a moving coil loudspeaker, at low ultrasonic frequencies a magnetostrictive rod, and at still higher frequencies, a piezoelectric or ferroelectric crystal. The transducer, whichever type is used, must be connected to the viscoelastic filament under investigation in such a way that longitudinal stress waves are propagated along the filament. In order to make measurements, the filament must be kept just taut, but the apparatus may also be Detector Driving Unit
Filoment
< -
I]-Fl-’I Oscillotor
Amplifier
FIG.5. Wave propagation apparatus for long filaments
H.Kolsky
332
conveniently used to study wave propagation in filaments with a large longitudinal prestrain. This may be achieved by prestretching it to any desired extent with a suitable loading mechanism. The detector is a probe of small mass, one end of which rests on the filament and the other end of which is attached to a piezoelectric crystal. The output from the piezoelectric crystal is amplified and fed to a cathode ray oscilloscope, where its amplitude and phase can be compared to a fixed signal from the driving unit. Thus measurements of the way luxl and 4 vary with the distance of the detector from the position of the driving unit may be determined. On the assumption that the material is linearly viscoelastic and that the reflections of the train of sinusoidal waves both from the detector and from the other end of the filament can be ignored, Eqs. (3.21) lead one to expect a linear plot of q5 against x, the gradient of which is w/c. In u, should also depend linearly on x with a gradient -a. For materials with high damping such as rubber, this is in fact found to be so, but for materials that have lower damping, such as nylon and polyethylene, the waves do not dampen out quickly enough, and damped standing wave patterns are observed along the
0
20
40
60
80
100
DISTANCE ( c m )
FIG.6. Phase angle
as a function of distance along filament.
Experiments in Solid Mechanics
333
DISTANCE (em)
FIG.7. Amplitude variation with distance..
filaments. When the filament is sufficiently long for the wave reflected from its end to be of negligible amplitude but where the reflection from the detector probe is still important, the form of these standing waves can be predicted if it is assumed that a constant fraction m of an incident wave is reflected at the probe. The two expressions corresponding to Eqs. (3.21) now become tan q5 - 1 + m exp( -2ax’) tan(ox’/c) - 1 - m exp( -2ctx‘)
(3.22)
and Uxr uo [1 - 2m
(1 - m) exp( -ax’) (3.23) exp( -ax’) cos(2ox’/c) + m2 exp( - 4cr~)J”~ ’
Experimental results for specimens of polyethylene that illustrate relations (3.22) and (3.23) are shown in Figs. 6 and 7. It may be seen that when ax >> 1, Eqs. (3.22) and (3.23) simplify to the two equations in (3.21). This type of technique has been used successfully by Ballou and Silverman (1944), Nolle (1947), and Hillier (1949). Another application of wave propagation methods that has proved useful in the frequency range of a few tens of hertz to a few kilohertz is to observe the free vibrations of unloaded rods of linear viscoelastic materials. Here a
334
H. Kolsky
system of standing waves are set up, and the lowest natural frequency in the case of a. clamped rod corresponds to a length A/4, where A is the wavelength of the waves along the rod. This applies to longitudinal and torsional vibrations. From the period of the vibrations the frequency is determined, and since in the fundamental mode the length is one-quarter of a wavelength the velocity of propagation can be found. Hence using Eq. (3.19) I E* I can be found for longitudinal oscillation or 1 G*I for torsional vibrations. With flexural vibrations the relevant elastic constant is Young’s modulus E and for viscoelastic materials the quantities 1 E*J and tan 6, are the relevant parameters. The analysis is, however, rather more complicated, but so long as the damping is not too high, i.e., << 1, the value of IE* 1 can be readily found from the period of flexural free vibration of a bar. If T is the period of free vibration of a bar of length 1 and density p clamped at one end and free at the other, K is the radius of gyration of a cross section about an axis in which the neutral axis cuts it, and the bar is assumed to be vibrating laterally in its fundamental mode, then the approximate relation for the parameter IE* 1 is
1 E* I = 4d14p/12.4~2T2.
(3.24)
When the bar is vibrating laterally in a higher mode the constant 12.4 is replaced by a higher number. For example, for the second clamped-free mode, 12.4 is replaced by 485.5. For longitudinal, torsional, or flexural vibrations the value of tan 6 can be found, so long as the damping is small, i.e., so long as 6 << 1, from the magnitude of the logarithmic decrement of the oscillations A . Thus as in the case of vibrations with an auxiliary inertia, the relation is Eq. (3.4). Similarly it is possible to obtain the viscoelastic parameters by observing the resonant response of a vibrating bar in forced oscillation. The peak frequency when 6 << 1 corresponds to the frequency of free vibration and the half-breadth ANIN 3: $ tan 6. Observations using the free vibrations of bars are described by Quimby ( 1 9 2 9 Wegel and Walther (1939, Randall et al., (1939), Parfitt (1949), Hillier (1951), and Benbow (1953). We will next consider how the mechanical response of various linear viscoelastic materials corresponds to the theoretical predictions of the simple types of model solid that were discussed in the previous section. The comparisons are shown in Fig. 8. Thus for a Maxwell type of solid, tan 6 was shown to be inversely proportional to the angular frequency, while for a Kelvin-Voigt solid it was found to be proportional to the frequency w. For the standard linear solid, tan 6 is proportional to the frequency at low frequencies and inversely proportional to w at high frequencies. Figure 8
Experiments in Solid Mechanics /
5i.,
/
4
-5
-4
-3
335
/ K
/ //
-2
-I
0
I
2
3
4
5
6
Log,, frequency
FIG. 8. Log tan6 variation with log n. A, Polyethylene; B, pmm; C, ebonite; D, polystyrene; E, PMF; F, GSP; M, Maxwell; K, Kelvin-Voigt; SL, standard linear solid.
shows the predictions of the behavior of log(tan 6 ) plotted against log n, where n is the frequency in hertz, for the three models. It should be noted that for this type of plot the Kelvin-Voigt model must be a straight line with slope + 1; changing the value of the parameters of the model only moves the plot to the left or right. Similarly, for the Maxwell model the slope must be - 1, although again the intercepts can be chosen arbitrarily; for the standard linear solid the plot must consist of a Kelvin-Voigt plot at low frequencies and a Maxwell plot at high frequencies, and only the position and height of the transition zone is affected by changing the parameters of the model. In the same figure, plots are shown of the measured values of tan 6 carried out by Lethersich (1950) for four typical polymeric solids in shear; these measurements were obtained for torsional deformation. It may be seen that the behavior of these polymers in no way corresponds to that of the models shown. As mentioned earlier, no simple model predicts behavior that corresponds to the response of most real viscoelastic solids, and in order to obtain better agreement with reality it is necessary to postulate a spectrum of relaxation times that results in linear operators P and Q of very high degree.
H. Kolsky
336
This precludes the analytical treatment of most boundary value problems in linear viscoelasticity. It may be seen from Figure 8 that for the four polymeric solids considered, namely, polyethylene, polymethylmethacrylate (pmm), ebonite, and polystyrene, a far better fit would be obtained on the assumption that tan 6 is independent of frequency over a wide frequency range. Ferry and Fitzgerald (1954)have shown from the theory of linear viscoelasticity that where the variation of El with frequency is comparatively slow, E,
N
+no dE,/do,
(3.25)
which may be written (see Hunter, 1960) d(ln El)/d In o N (2/n) tan 6.
(3.26)
Equation (3.26) may be integrated to give the following relation over a frequency range wo --* o,where tan 6 is effectively constant: = (2/n) tan 6 ln(w/oo), ~n[E,(~)/E,(oo)]
or, since tan 6 is assumed to be very small compared to unity, El(o) 2 E,(oo)[l
+ (2/7c) tan 6 ln(o/oo)]
(3.27)
(see Kolsky, 1956). If this relation is a reasonable approximation to the frequency dependence of El(o), and 6 << 1, we may use the approximate relation
I I
c(w) = ( E* /p)’/’ sec 46 2
N
(El / p ) ’ I 2
c(oo)[l + (tan 6/71) In(w/oo)].
(3.28)
Equation (3.28) provides a useful relation in the later discussion of pulse propagation in specimens of high polymers. It should perhaps be mentioned here that whereas polymeric materials, which are composed of giant long-chain molecules with many degrees of freedom, never even approximately behave like simple model solids, a number of resins of very much lower molecular weight can approximate Maxwell solids quite closely in their behavior. The results of measurements on two such “ organic glasses,’’ namely,
2-hydroxy-2,4,4,6,5’-pentame thylflavan (PMF) and glycerol sextol phthallate (GSP) over the range to lo3 Hz are also shown in Fig. 8. It may be seen that for both of these materials there is a fairly wide range of frequency over which their mechanical behavior corresponds reasonably closely to that of a Maxwell solid. Thus for PMF
Experiments in Solid Mechanics
-- --
337
there is a linear decrease of log tan 6 from to 100 Hz, while for GSP the linear region appears to extend from to lo-’ Hz. If we know the value of the propagation constants c and a for trains of sinusoidal waves propagated along a linear viscoelastic solid over a sufficiently wide frequency range, since linear superposition obtains for such materials we can predict the way a pulse of any shape will change as it travels through such a solid. All we need to do is to Fourier analyze the initial pulse shape and then Fourier synthesize the later shapes with the aid of the values of c and a. Thus, if the pulse at the origin x = 0 is represented by W
A(@) exp(iot) dw,
~(0, t )= 0
where A ( o ) is in general complex, then the pulse shape after the pulse has traveled a distance x will be represented by E(X,
t )=
5
m
A(w) exp( - a x ) exp iw(t - x/c) dw,
(3.29)
0
and if the values of a and c are known over the relevant frequency range, E ( X , t) can be determined. In practice, a Fourier sum is substituted. for the Fourier integral and we may write n
+
~(0,t ) = C a, cos mot 0
n
b, sin mot,
(3.30)
1
and having set a basic time interval 2 n / 0 0 we find the values of the an’sand the b,’s by Fourier analysis. We then have for the pulse at a distance x l , &(xlt ) = C a, exp( - a , x l ) cos mo(t- x 1/c,) 0
n
+ C b,
exp(-a,x,)
sin mo(t- xl/c,),
(3.31 )
1
where a, and c, are the attenuation constant and the phase velocity of propagation, respectively, of a train of sinusoidal waves of frequency m o. [In practice, it is helpful to use an expression x l ( l/c, - l/c’)instead of xt /c, in Eq. (3.31)in order to keep the pulse in the basic time interval 2 z / w 0 . c’ is the “pulse” velocity, it is a constant for the summation, and its choice is not at all critical, being somewhere between c1 and c,.] Figure 9 shows an apparatus that has been used by the author (Kolsky, 1956) to observe the change in shape of a stress pulse as it travels along a
H.Kolsky
338 * ,
SUSPENSIONS
EXPLOSIVE
MICROMETER HEAD
HEATING WIRE PHOTOCELL
-1
I
OSCILLATOR
CATHODE RAY
. OSCILLOGRAPH
AMPLIFIER
FIG. 9. Apparatus for pulse propagation in rods.
freely suspended rod of a linearly viscoelastic material. The initial pulse is produced by the detonation of a small explosive charge of lead azide at one end of the rod specimen. When the pulse arrives at the other end of the rod, the displacement produced by it is detected by a capacitor arrangement that consists of a grounded plate, which is the silvered end of the rod, and an insulated plate, which is a metal disk mounted on a micrometer head. This disk is set parallel to the flat end of the rod and any desired gap between the end of the rod and the insulated plate can be achieved by adjusting the micrometer head. It is important to ensure that the apparatus is not subject to drafts and that the bar is not swinging, so that the capacity between the rod and the insulated plate remains constant in the absence of a stress pulse. Any movement of the end of the bar results in a change in the capacity between it and the insulated plate, and by charging the insulated plate to a high voltage (- 500 V) through a high resistance (- 10 MR) any change in capacity is converted into a voltage change, which can be amplified and fed onto the deflector plates of a cathode-ray oscillograph. Since the capacity of the unit is very small indeed ( - 1 pf) the time constant is too short for reliable observations to be made, and a ballast capacitor of 10,OOO pf was connected across it. This reduced the sensitivity considerably, but the setup was still quite adequate for reliable measurements to be made. In order to trigger the oscillograph, the flash from the explosion of the lead azide charge was utilized, and it fell on a photocell, which provided a triggering pulse. So long as the displacement produced by the pulse is small compared with the capacitor gap, the voltage output from the unit is proportional to the displacement, and the oscillograph record shows a steplike rise in the trace when the pulse arrives at the end and is reflected. To obtain the pulse shape
.
Experiments in Solid Mechanics
339 -
10
- 9 - 8
-7 -6
TIME (microseconds)
FIG. 10. Particle velocity profiles of pulse after different distances of travel through pmm rods. (A) 1.2, (B)3, (C) 6, (D) 8.4 m.
in terms of particle velocity this trace was differentiated,and Fig. 10 shows how the particle velocity pulse flattens and extends as it travels increasing distances along polymethylmethacrylate rods. The analysis of pulse propagation for the shape of a strain pulse given in equations (3.30) and (3.31) applies equally if the stress c or the particle velocity I/ associated with the pulse is considered. Figure 11 shows a comparison between the observed shape of the particle velocity ofa pulse after it had traveled through 60 cm of polyethylene rod and the predicted shape using the values of c and 01 measured by Hillier (1949) for similar samples of the material at frequencies of the relevant Fourier components. In making this comparison it was found sufficient to assume that the pulse at the origin, i.e., at the end where the explosive charge was situated, was sufficiently short in duration to be considered a Dirac 6 function, i.e., a, = const for all n and b, = 0 for all n in Eqs. (3.30) and (3.31). It can be seen that agreement between theory and experiment is highly satisfactory. Comparisons were also made by Fourier analyzing the pulse shape after it had traveled a distance x1 and then synthesizing the predicted shape for some greater distance x 2 . Agreement here was also found to be extremely satisfactory.
H.Kolsky
340
0
I00
300
200
500
4 00
TIME (mlcroscconds)
FIG. 11. Comparison of experiment with theory (polyethylene rod 60 cm long). -, Observed pulse; x , points obtained by Fourier synthesis.
It was observed experimentally that the pulse shapes for different distances of travel in polymer rods were all geometrically similar in the sense that if the particle velocity V(x, t) for one distance of travel x1 could be expressed as q x 1 , t) =f(t),
(3.32)
then at some other distance of travel x2 the observed shape could be written (3.33) This was found to be true not only for different distances of travel in rods of one polymer, but for pulses traveling in different polymers, and it seemed highly unlikely that this observation was purely fortuitous. It seemed possible that it was related to the fact that tan 6 did not change very much with frequency for any of the polymers studied. The pulse shape expected after a distance of travel x1 is given from (3.29) as
where it is assumed that at the origin (x = 0) the pulse can be represented by a Dirac 6 function, i.e., V(0, t ) =
5
m
0
A cos wt do,
Experiments in Solid Mechanics
34 1
and that the frame of reference is traveling at a velocity c‘ with the pulse. From Equation (3.19) we see U(W)
= o tan 6/2c(o),
and since c(w) varies only very slowly with o and tan 6 is assumed to be constant, a reasonable approximation is a(o)= Lo, where L is taken to be constant. Thus (3.33) takes the form m
V(x,,
t)=
A exp(-lwx,) cos o ( t - xl(l/c - l/c’)) d o ,
(3.35)
0
1 A exp(-Lmwxl) cos mw(t/rn - x,(l/c - l/c’)) d ( m o ) , (3.36) m
V(x2, t) =
0
where m = x2/x1. Equation (3.36) has the same relation to (3.35) as (3.33) has to (3.32), so long as (l/c - l/c’) is itself either constant or is a function of ox. As shown in Eq. (3.28) for a material for which 6 << 1 and tan 6 is independent of o,
c(0’) 2 c(oo)(l + (tan 6/n) In(w/oo)). To the same degree of approximation, we have l/c(o) N ( l/c(oo))(1 - (tan 8 / n ) ln(o/oo)),
so that if we choose c’ such that l/c’ = ( l/c(wo))(1
+ (tan S/n)ln(x/xo)),
(3.37)
where xo is a disposable constant, we get (l/c(o) - l/c’) = [tan 6/nc(oo)]In ox/ooxo .
(3.38)
Equation (3.37) implies that the pulse velocity c’ is not quite constant, but decreases very slowly with increasing distance of travel. If we now substitute for (l/c(o) - l/c’) in Eq. (3.36) from (3.38), we find that the material properties occur only in the form of the parameter x1 tan S/co and the observed result of the same pulse shape for rods of different materials is therefore not unexpected in view of the fact that for all of them tan 6 does not vary very much with frequency and is small compared with unity. A Fourier synthesis of the “universal” pulse shape is shown in Fig. 12, as well as the experimentally observed pulse shape for a pulse that had traveled through 6 m of pmm. An empirical fit was achieved on the assumption that x tan S/c(wo)was 100 pec. It can be seen that the agreement is extremely good.
H . Kolsky
342 10
'-
98-
In
c
7-
3
x L
!?
6-
.-
+
-: w
5-
P
'= r
4-
_I
a
5
32I -
TIME
FIG. 12. Universal pulse shape compared with experimental pulse shape for 6-m prnm rod. x , experimental.
0, calculated;
The value for c(wo)can be determined from the time of transit of the pulse and was found to be about 2300 m/sec, so that the value of 100 psec for x tan S/c(oo) would correspond to a value of tan 6 of about 0.04. This is in reasonable agreement with that measured by Lethersich for torsional oscillations of pmm at comparable frequencies. We have so far considered only linear viscoelastic waves in rods, but in order to be able to treat general problems of wave propagation in three dimensions in linear viscoelastic solids, we also need to know the dynamic mechanical response of linear viscoelastic solids to volume changes. The measurement of such volume viscosity effects experimentally is by no means straightforward, since direct measurements are difficult to carry out and involve errors related to the presence of voids in the material, while indirect determination, e.g., the use of measurements of extensional and torsional response, are notoriously inaccurate since they involve calculating small differences between two large quantities. Lifshitz and the author (Lifshitz
Experiments in Solid Mechanics
343
HEATING WIRE EXPLOSIVE
CONDENSER MICROMETER HEAD
FIG. 13. Apparatus for measuring spherically diverging pulses in blocks.
and Kolsky, 1965) have carried out direct measurements on the propagation of spherically diverging pulses in blocks of plastics. The apparatus that was used is shown diagrammatically in Fig. 13. The relevant propagation constants for P waves is k + $G,where k is the bulk modulus of the solid. As in the experiments on rods, the pulses were produced by the detonation of small explosive charges and the pulses that arrived at the opposite face of the cylinder were detected with capacitor units. Blocks of polyethylene, pmm, and polystyrene were employed, and the observed pulse shapes were compared with those predicted on the basis of three theoretical assumptions about the nature of volume viscosity. The first assumption (A) was that tan 6 for volume changes (tan 6,) was equal to tan 6, for shear. This assumption was first made by Quimby (1925) on the basis that what appears as a purely volume change in a polymer macroscopically involves, because of the complicated molecular structure, both volume and shear changes on a molecular scale. The second (B), which may be termed the Stokes assumption, is that volume changes are essentially elastic and tan 6 for such strains is effectively zero. The observed pulse shapes were found to lie between these two extreme assumptions and it was found both for polyethylene and for pmm that assumption C, [tan 61, = [0.2 tan d],, led to remarkably good agreement with the observed behavior. The three curves and the experimental results for polyethylene are shown in Fig. 14. The results for polystyrene were less clear-cut than those for polyethylene and pmm since the damping in this material is rather low. However, the experimental results did show that Assumption C was as good as any in predicting the pulse shapes.
H.Kolsky
344
14
12
10
6
8
4
2
0
p stc
FIG.14. Comparison of theory (-) and experiment ( 0 )for observed pulse shapes in polyethylene block. tan 6, = 0.0 (A), 0.023(B), 0.110 = tan 6, (C).
The fact that the velocity of propagation of trains of sinusoidal waves increases with their frequency results in an interesting reflection effect, which was first predicted by Kolsky and Lee (1962).The basis of this effect is that if a longitudinal elastic pulse is traveling in a medium that has an acoustic impedance Zl (= p1 c l , where p1 is the density and cI the velocity of propagation in this medium) and it is reflected at the boundary with a second medium of acoustic impedance Z2 (= p 2 c 2 ) then if Zl > Z2 the reflected pulse is of opposite sign to the incident pulse, i.e., a compressive pulse is reflected as a tensile pulse and vice versa. Conversely, if Z , > Z1 then the reflected pulse is of the same sign as the incident pulse. When 2, = 2, there is no reflected pulse. If the ratio of the amplitudes of the incident and reflected pulses is denoted by R, we have
When one or both of the media is viscoelastic, relation (3.39) cannot be applied to the pulse as a whole. If, however, a train of sinusoidal waves u = uo exp iot travels in a linearly viscoelastic medium [which now has a complex acoustic impedance 2, = pl(C, + iC,)]and it is reflected at the boundary with a second linearly viscoelastic medium that has a complex acoustic impedance 2, = p2(C1 + iC2), Eq. (3.39) can still be applied, except that of course-now R is, in general, complex.
Experiments in Solid Mechanics
345
If we want to predict the shape of the reflected pulse when a pulse of arbitrary shape is reflected at the boundary between two media (at least one of which is linearly viscoelastic)we musf Fourier analyze the incident pulse, then apply Eq. (3.39) to each of the Fourier components, and finally synthesize the reflected pulse shape. In general, this results in the shape of the reflected pulse differing only slightly from that of the incident pulse. However, when the value of 12, I crosses the value of 12, I at some frequency w, that is in the relevant frequency range of the Fourier spectrum of the pulse, what has been termed a crossouer effect is observed. For example, if a dc pulse of compression is propagated along an elastic rod that has an elastic acoustic impact Z , and the end of which is in contact with a rod of linearly viscoelastic material that has a complex acoustic impedance 2, and the value of I Z 2I crosses the value of Z , at some frequency a,[i.e., for w < w c , Z , > I 2, 1 and for w > w,, 2, < I Z, I] then the dc pulse is reflected as an S-shaped pulse with a compressive front portion and a tensile back portion. This was the prediction originally made by Kolsky and Lee (1962). Recently, Moffett (1971, 1973) carried out experiments that verified this predicted effect. Moffett used as his two viscoelastic materials polystyrene and unplasticized polyvinyl chloride (PVC). The former is almost elastic, while the latter is fairly lossy. The stress pulses were produced by the axial impact of a short polystyrene pellet at the free end of the polystyrene bar and the pulse shapes were observed with strain gages affixed to it. As the temperature was varied from room temperature (20°C)to about 55"C,the shape of the reflected pulse was shown to change from essentially compressive to purely tensile. At intermediate temperatures the cross-over effect could be observed with the reflected pulse consisting of a compressive region followed by a tensile one.
IV. Tensile Shock Waves in Rubber So far we have discussed the mechanical response of solids, which is linear in the sense that although the stress-strain relation may be time dependent, the form of the relation is not amplitude dependent. We now come to the problem of the mechanical response of solids under conditions of loading where the stress-strain relation is nonlinear. Under these conditions, one of two types of wave may be propagated through the solid. Thus, if the material yields, i.e., if the tangent modulus decreases with increasing stress, plastic waves are propagated through it. If instead it becomes stiffer,i.e., the tangent modulus increases with increasing stress, shock waves are propagated. Here
H.Kolsky
346
we will discuss first some recent experimental work on tensile shock wave formation in stretched natural rubber carried out by the author. The formation of compressive shock waves in solids is a well-investigated field of study and the reader is referred to Duvall (1966) for a review of the experimental and theoretical work on this subject. The basis of such shock wave formation in solids is that the compressibility of solids increases with increasing hydrostatic pressure, so that the propagation constant K + 4G/3 increases with increasing wave amplitude (cf. Bridgman, 1931). The formation of tensile shock waves does not appear to have received much experimental attention, although Lee and Tupper (1956) have discussed the possibility of the formation of such tensile shocks in stretched steel wires. In the description of the experimental observations of longitudinal sinusoidal waves along filaments of rubbers and high polymers, it was pointed out that the apparatus shown in Fig. 5 could be used to study the propagation of small-amplitude sinusoidal waves along prestretched filaments of such materials. Experimental investigations of this type were carried out by Hillier and Kolsky (1949) and Hillier (1950) on filaments of natural and synthetic rubbers, and it was found that as the material was stretched the velocity of propagation of sinusoidal waves c increased very markedly at the same time as the attenuation constant a decreased. The results for natural rubber are shown in Fig. 15, where c and a are plotted as functions of the prestretch ratio A. It may be seen from the figure that at a stretch ratio of 4, c has increased to more than six times its value for unstretched rubber (A = 1) and a has fallen to a few percent of its value for the unstretched state. These measurements would indicate that if rubber is first prestretched to several times its original length and an incremental tensile strain pulse is then propagated along it, the incremental pulse will rapidly develop a shock front. (It should be noted that from the point of view of rate of increase of c with A, this is sufficiently high, even when A = 2, for tensile shock waves to develop. Unfortunately, the attenuation is still too large for shock fronts to persist and a value of A of about 4.5 is required before such shock fronts can be observed.) An apparatus that has been used by the author (Kolsky, 1969, 1972) is shown in Fig. 16. The rubber specimen employed was a vulcanized gum stock of square cross section x f in.). In a typical experiment, a length of the rubber was stretched to about five times its original length by means of an electrical winch, and the total length was then about 13 ft. A short section of the band at one end, about 10 in. in length, was then stretched to a length of about 11 in. Thus the stretch ratio in this section was about 5.5. The additional stretch in this section was held by a short length ofpiano wire and the incremental tensile pulse in the rubber was generated by volatilizing the piano wire electrically.
(4
Experiments in Solid Mechanics
EXTENSION RATIO
341
A
FIG. 15. Phase velocity (V)and attenuation ( x ) of 500-Hz sinusoidal waves in natural rubber as a function of stretch ratio.
1 stretching machine
L-auIIiliary stretcher
\rJ
FIG.16. Apparatus for observing tensile shock waves in stretched rubber bands.
348
H.Kolsky
In order to observe the change in shape of the pulse as it traveled along the rubber band, a number of straight wires were fixed transversely to the length of the band and were placed in uniform magnetic fields so that when the pulse arrived they cut the lines of magnetic force. Thus the voltage outputs from these wires were proportional to the particle velocities produced by the pulse. This technique was first devised by Ramberg and Irwin (1957) and later developed by Ripperger and Yeakley (1963). It is extremely convenient when large displacements take place dynamically.
2
-
u 10m sec
FIG.17. Development of tensile shock waves in stretched rubber specimens.
Figure 17 shows how the shape of the pulse changed as it progressed along the rubber band. It can be seen that after it had traveled about 7 ft, a definite shock front developed. More recently, " compression pulses " have been propagated along stretched rubber specimens. In this work the strain in the end section was made lower than in the rest of the specimen and this region of lower tensile strain was held by the piano wire. When the wire was
Experiments in Solid Mechanics
349
volatilized a region of lower tensile strain propagated along the band. As can be seen from Fig. 15 the velocities associated with lower stretch ratios are lower than those at higher stretch ratios; thus the velocity of the “peak” of this pulse is lower than that of the rest of the pulse and instead of a tensile shock front being generated a shock tail is formed at the rear end of the pulse. V. Plastic Wave Propagation
The earliest treatment of plastic wave propagation was formulated by Donnell (1930). Donnell considered the propagation of tensile waves along a wire that has a bilinear stress-strain relation of the form illustrated by
FIG. 18. Bilinear stress-strain curve.
Fig. 18. It is assumed that up to a stress doand strain the wire behaves in a linear elastic manner, i.e., 0 = EE. At a stress do the wire yields, and for stresses greater than no the stress-strain relation follows a second linear portion, which has a gradient E. Thus = EE
and
do = E c O ,
for
0
< do,
and (0
- do) = E ( E- eO),
so that d = do
for
+ E(E - E o ) .
CT
> do,
H.Kolsky
350
In the elastic region E < E, the stress-strain relation for unloading is the same as it is for loading. In the plastic region, however, the material unloads elastically, so that when the wire is completely unloaded from a strain E > E , , a permanent strain of magnitude el remains, where = [(E - E‘)/E] ( E - E,). Donne11 showed that if one end of such a wire is suddenly stretched at a uniform velocity V, a single step wave is propagated along it, so long as V < c, E, [c, = (E/p)’”, the velocity of extensional elastic waves along the wire, and p is the density of the wire]. The strain behind the wave front, which is traveling at a velocity c,, is given by V/c, and the stress by p c , V . When the end of the wire is stretched at a velocity V > c, E, , however, two wave fronts are propagated along the wire. The first is an elastic-wave front of stress amplitude a,, which travels at a velocity c, .This is followed by the plastic-wave front, which travels at a velocity c’ = (E‘/p)’’2and has a stress amplitude IJ = a,(l - c’/c,) + pc’V. The two fronts are shown in Fig. 19.
1 -t
FIG. 19. Plastic waves in bilinear material.
After Donnell’s paper in 1930, the subject received little attention until the Second World War, when more general theories were developed by Taylor (1946), von Karman and Duwez (1950), and Rakhmatulin (1945). These treatments all show that when the end of a long metal wire that has a linear elastic region is stretched at a constant velocity V, so long as V < a,/pco, an elastic-wave front of stress amplitude pc, V is propagated along it at the velocity c, = (E/p)’”. When, however, V > no/pco two wave fronts are generated. First there is an elastic-wave front of stress amplitude I J ~behind which the stress rises
Experiments in Solid Mechanics
35 1
steadily up to the stress ol,which is where the plastic-wave front commences. The plastic-wave front travels at a velocity c1 = S1/pl/’, S , being the tangent modulus of the stress-strain curve at the stress ol.The relation between the strain E~ and V turns out to be
[S is here the value of (do/dE) at the strain E ] . In the region between the two wave fronts there is a gradual increase in o and E . The stress distribution in this region continually changes shape with time. Each part of the pattern where the stress is of and the strain E’ travels forward with velocity c‘, where, c’ = (S’/p)’’’ and S’ = (do/&), at the point on the stress-strain curve for increasing stress, where the stress is o’and the strain E‘. The treatment described above is valid for small strains, when the Lagrangian and the Eulerian coordinate systems predict essentially the same behavior. For large strains the treatment as it stands is valid only for a Lagrangian coordinate system; then o1 is the engineering stress and not the true stress. The equations can be readily converted into Eulerian coordinates (see, e.g., Kolsky, 1953; Abramson et al., 1958; Goldsmith, 1960). The predictions of the theory of plastic-wave propagation were tested experimentally by Duwez and Clark (1947), who examined the strain distribution in copper wires that had been impulsively stretched by a falling weight. The results showed that a plastic-wave front was in fact propagated along the wire and its magnitude corresponded well with that predicted by Eq. (5.1). The strain distribution ahead of this wave front was not, however, in good agreement with that predicted by the theory. This was at first attributed to the complicated unloading waves that were set up in the experiments when the stress at the end fell to zero. However, as Lee (1956) pointed out this could not be the whole story, and he suggested that the observed differences were due to the sensitivity of the stress-strain relation to the rate of loading. Malvern (1951) investigated the type of strain distribution that might be expected if the metal were assumed to be slightly rate sensitive. Figure 20 shows the strain distributions expected from the rate-independent theory and the rate-dependent theory as recently calculated by Malvern (1965). This type of strain distribution is much closer to that observed experimentally. Another consequence of the rate-independent theory of plastic-wave propagation is that if a material is stretched quasi-statically up to some strain value E‘ and an incremental tensile stress AE is then applied, this incremental strain should travel at a velocity c‘ = (S‘/p)’/’, which is in general very much
H.Kolsky
352
0
10
I
I
20
30
40
X (in.)
FIG.20. Malvern's calculations of plastic-wave profiles. --, --, rate-independent solution.
Rate-dependent solution;
lower than the elastic-wave speed co = (E/p)'I2. Bell (1951) carried out such an experiment on a steel wire while it was being drawn, and found that the incremental pulse traveled at a velocity co, which was very much higher than c', the plastic velocity in the wire at that strain. This result was later confirmed for copper by Sternglass and Stuart (1953) and for lead by Alter and Curtis (1956). The importance of rate dependence in the propagation of plastic-wave propagation has thus been firmly established and experimental work continues to elucidate its various manifestations. In his original work on plastic-wave propagation, Bell (1951) used electrical strain gages to measure the wave shapes. Soon after this, however, he developed an extremely elegant optical technique for observing stress wave propagation and the dynamic stress-strain response of materials (Bell, 1956). The principle of this method is to rule a diffraction grating on the specimen and to observe the position of the diffracted image produced by it. When the strain in the specimen changes, there is a corresponding change in the number of lines per inch in the grating, and this changes the angle that the diffracted image makes with the incident beam. Bell allowed this diffracted beam to fall on a V-slit behind which there was a photomultiplier cell, the output of which was fed onto a cathode-ray oscillograph. Unfortunately this output would change not only when the strain in the specimen changed but also when the grating surface rotated as a result of the passage of the stress wave. To allow for this, Bell observed the specularly reflected beam with a second V-slit, and was thus able to separate rotations from changes in strain. The apparatus is shown diagrammatically in Fig. 21. With this type of apparatus Bell has been able to make subsequent measurements of the effects of rate of strain on various metals (Bell, 1964, 1965). Other
Experiments in Solid Mechanics
353
FIG.21. Bell's apparatus using diffraction gratings for measuring dynamic strains.
experimental work in this field to which reference should be made has been carried out by a number of workers, and the reader is referred to the proceedings of a symposium on the subject (Lindholm, 1968), as well as to the work of Campbell (1973) and to an investigation by Douch and the author (Kolsky and Douch, 1962). As mentioned earlier in this article, the direct measurement of the stressstrain relation of materials at high rates of loading is made very difficult by the fact that as the loading rate increases the inertia of first the loading mechanism and finally the specimen itself introduces stresses that are not related to the constitutive relation of the material, but only to its density, and that eventually the experimenter is faced with a problem in wave propagation whether he wants it or not. Some time ago the author devised an apparatus that he called the split Hopkinson bar, which attempted to overcome some of the usual difficulties involved in stress-strain measurements at very high rates of straining (Kolsky, 1949). This technique employs elastic waves to produce the deformation and to observe the response of the sample. Although the method suffers from some serious defects, it has been used up to the present for making such measurements (see, e.g., Davis and Hunter, 1963; Jahsman, 1971). The arrangement is illustrated in Fig. 22. The specimen, which is in the form of a short cylinder, is placed between two steel bars of the same diameter as the specimen. A compression pulse is sent down the main bar either as a result of the impact of a projectile at the end of one of the bars or as a result of the detonation of a small explosive charge. The elastic waves set up in the two bars are recorded on a cathode-ray oscillograph and from the
354
H.Kolsky INERTIA
CYLINDRICAL CAPACITOR
AMPLIFIER I
PARALLEL PLATE CAPACl TOR
AMPLIFIER 2
CATHODE
0 SClL LOGRAPH
FIG.22. Split Hopkinson bar apparatus for measuring dynamic stress-strain curves.
outputs the stress-time and the strain-time curves for the specimen are obtained, and hence the stress-strain curve derived. The advantages of the method are that it can be used for very high rates of straining and that the effects of the inertia of the bars can be taken into account exactly. The disadvantages are that unless the specimen is very short the stress distribution along it will not be uniform at high rates of strain. However, when it is very short, friction between the ends of it and the faces of the two bars distorts the radial stress distribution. This puts a serious limitation on the highest rates of straining for which it can be meaningfully applied. An additional difficulty is associated with the radial inertia of the specimen, the effect of which depends on the time derivative of the strain rate. In spite of these limitations, the method has produced some interesting experimental data.
VI. Experimental Studies in Dynamic Fracture
The fracture of brittle solids is in general a catastrophic phenomenon and the study of the dynamics of the problem has been a subject ofexperimental concern for well over a century. Since the propagation of brittle fractures in solids, once they have started, is an extremely rapid phenomenon it is not surprising that there are rapid changes in the stress field surrounding the propagating crack. Hence the connection between dynamic fracture and stress-wave propagation is a very close one.
Experiments in Solid Mechanics
355
Some of the earliest experimental work in this field was carried out by John Hopkinson (1872),who studied the strength of steel wires under impulsive tensile loads. A ball-shaped weight pierced by a hole was threaded on the taut vertical wire and was dropped down to hit a light clamp attached to the bottom end. Hopkinson showed that under these conditions of loading if the height of fall was gradually increased, the wire first broke at its upper support, and that the ability to break the wire did not, over a large range of weights, depend on the size of the weight but only on the height from which it dropped. He further showed that when the height was four or more times the minimum at which fracture first took place at the top of the wire, fractures occurred at the lower end. All these results are explicable once it is realized that the fracture takes place as a result of the tensile elastic wave that is propagated along the wire. Thus if the velocity of impact is V and the mass of the clamp is neglected, the stress set up at the bottom end of the wire is pco V,where p is the density and c0 = (E/ p )l ’ ZIn . order to build up twice the tensile stress, the velocity Vmust be doubled, and thus in order to achieve the same stress without reflection of the pulse the mass M must be dropped from four times the height at which fracture occurred at the top of the wire. Little more experimental work on dynamic fracture appears to have been carried out until Bertram Hopkinson, the son of John, investigated the phenomenon of spalling, which is a familiar one in a military context. When a rapid impact occurs at one face of a metal plate, only plastic deformation occurs around the region of impact, while a large spa11 (or scab) is shot off from the opposite face. B. Hopkinson (1912) explained this phenomenon as being the result of the reflection of the compression pulse set up by the impact at the free surface of the plate. The type of stress distribution that might be expected for the reflection of a compression pulse at normal incidence is shown in Fig. 23. The condition that the normal stress vanishes implies that the reflected pulse is equal in magnitude and opposite in sign to the incident one, and the six stages of the reflection process show how tension is built up inside the material although the boundary is stress free. It can be seen that for a pulse of the shape postulated in the figure the maximum tension is first observed at a distance of a little less than half the pulse length away from the free surface. In the impact of a projectile on the surface of a plate, a spherically diverging stress wave is set up and only directly beneath the point of impact is the incidence normal. When an elastic wave is reflected at angles of incidence other than normal, two waves are in general reflected (Kolsky, 1953), and the stress distribution is complicated considerably. However, the general qualitative behavior is not unlike the situation illustrated in Fig. 24, where the two surfaces of the plate are denoted AB and CD. The impact is assumed
356
H.Kolsky
I
I
I ,'-' \
FIG.23. Reflection of compression pulse at free boundary.
to be at the point P and the reflected tensile wave assumed to diverge from the image point P' of P in the surface CD. Apart from the effects of oblique incidence, it may be seen that as we diverge from the axis PP' the distance of travel of the wave increases, and hence its amplitude decreases. The region where the amplitude is large enough for spalls to be formed is thus limited. Now along the axis PP' the amplitude of the compressive pulse is very large. When reflected, it produces a crack parallel to the free surface and very close to it. The rest of the compressive pulse is now reflected at this new free surface and the process is repeated a number of times so that there are a series of parallel cracks
C
D L..SPALL
P' FIG.24. Spa11 formation in large plates.
Experiments in Solid Mechanics
357
perpendicular to the axis PP'. In contrast, at the edge of the spall the amplitude is sufficient to produce only one crack, so that the shape of the region of spalling is approximately that shown in the figure. In the spalling phenomenon just described, it was assumed that only the waves reflected from the bottom face of the plate were intense enough to fracture the specimen, and when the specimens are in the form of large plates only a single spall is in fact observed. With smaller specimens, however, pulses reflected from the sides of the plate can produce their own spalls, and interference between pairs of reflected pulses can result in planes of reinforcement where fractures are observed. The effect is illustrated in Fig. 25,
-pg-J-A
e
P
/
D
/ c
FIG.25. Reinforcement fractures in small rectangular specimens.
which represents the cross section of a small rectangular specimen. The explosive charge is assumed to be at P in the top surface AB; BC and AD represent the sections of the side surfaces, and DC a section of the bottom surface. The compression pulse diverging from P is reflected from AD, DC, and C B as three spherically diverging tensile pulses that have P,, P,, and P , as their respective image points. Reinforcements can occur between these pulses. For example, the pulses reflected from DC and BC reinforce in the plane containing C Y. which is perpendicular to P , P,, and C Y is in fact where a fracture plane is observed (Kolsky and Shearman, 1949; Kolsky, 1953; Kolsky and Rader, 1968). A similar fracture is observed in the plane through D perpendicular to PI P 3 , this is produced by the reinforcements of the tensile pulses reflected from AD and DC. Additional reinforcements occur when a cylindrical specimen is employed instead of a plate, and the situation is illustrated in Fig. 26. The explosive charge is assumed to be at the center of one of the flat end faces P. A spherical compression pulse will diverge from P and will be reflected by the cylindrical and the bottom faces of the specimen. If the amplitude of the tensile pulse reflected from the opposite flat face of the cylinder is sufficiently
358
H . Kolsky S
P
T
B
A
G
FIG.26. Reinforcements in cylindrical specimens.
high, a plane spall (denoted K H ) will be formed perpendicular to the axis of the cylinder. Similarly, the tensile wave reflected from the cylindrical surface produces a spall on the top surface in the form of a circular crack, which is denoted S and T. This pulse reflected from the cylindrical surface converges onto the axis of the cylinder so that the region around this axis is subjected to a large tensile stress, and a region of intense fracture, which is concentrated in a narrow cylindrical pencil around the axis, is observed. This is denoted PC in the diagram, and it can be seen that wavelets diverging from P and reflected at points A and B arrive simultaneously at a point C along the line. The other fractures observed are a reinforcement region in the shape of a cone, which arises from the interference of the pulse reflected at the cylindrical surface with the pulse reflected from the bottom face. This surface is denoted L and M in the figure. Finally, there is a region of intense damage around P, which is caused by the high compressive stresses around the seat of the explosion and which is produced by hoop-tensile stresses. All these phenomena have been observed experimentally with cylindrical specimens of plastics, the different effects predominating at different length to radius ratios of the cylinders (see Kolsky and Shearman, 1949; Kolsky, 1953; Kolsky and Rader, 1968). All the spalling phenomena discussed above are explicable in terms of the specular reflection of stress pulses at the free boundaries of specimens. There
Experiments in Solid Mechanics
359
are, however, a number of fracture phenomena that are produced by intense stress pulses and result when the length of the pulse is comparable with the linear dimensions of the specimen, and these cannot be explained in terms of simple reflections. If we make an optical analogy the situation is one of physical optics rather than of geometrical optics, and it is the difffaction of the pulse and not the rejection that is producing the tensile stress, which results in fracture. A rather striking example of this type of behavior is a phenomenon, first reported by Kolsky and Shearman (1949), of the fracture of a cone tip when an explosive charge is detonated at the center of fhe flat base of a long conical specimen. The compression pulse traveling down the cone builds up a tensile tail and this results in a fracture some little distance from the apex of the cone. Similar types of fractures are observed close to the corners of plane specimens where the cylindrically diverging waves enter regions of decreasing cross sections (Kolsky, 1953). Similar types of fractures in metal specimens are described and discussed in the book by Rinehart and Pearson (1954). Another fracture phenomenon that is directly related to stress wave propagation is the limiting velocity of propagation of running cracks in brittle solids. It is found experimentallythat however large the tensile stress applied to a specimen may be, brittle cracks never travel at velocities higher than about one-half the velocity of shear waves, which for most brittle solids is about one-third the velocity of dilatational waves. It is clear that so long as the stresses built up in finite regions of the specimen are small enough for the deformation to be elastic, a running fracture cannot propagate at a velocity in excess of the velocity of dilatational waves since this is the highest signal velocity in an elastic medium. It could also be plausibly argued that since the stress field in which a fracture propagates is not a simple one, the arrival of a shear wave might be necessary before fracture can progress, but it is difficult to see why the limiting fracture velocity should be only about one-half this lower limit. The most plausible explanation of the observed value of the limiting velocity is due to Yoffe (1951), who showed that if we consider the propagation of a running crack in a uniaxial stress field we find that so long as the velocity of crack propagation is small compared with the velocity of elastic waves, the hoop tension in the region of material just in front of the crack tip has its maximum perpendicular to the direction in which the fracture is progressing, so that there is no tendency for the fracture to change its direction. When, however, the velocity of crack propagation reaches about onehalf the velocity of shear waves, the maximum value of begins to deviate from the direction along which the crack is traveling, and there are two maxima making equal angles with this direction. Under these conditions the
360
H.Kolsky
propagation becomes unstable, and there is a tendency to bifurcation of the fracture or to an erratic path, which results in a hackle structure of the fracture surface, in contrast to the mirrorlike surface that occurs at lower velocities. Such bifurcation is often observed experimentally as is the formation of hackle surfaces when brittle solids are subjected to large tensile stresses. The experiments of Christie (1952) support the Yoffe treatment of the problem. He has taken high-speed photographs of the progress of a running crack in a glass plate under conditions where the stress could be observed photoelastically. When such a running crack suddenly enters a region of high tensile stress, it is found to bifurcate, and the fracture surface, which before bifurcation occurred was smooth and mirrorlike, became rough and hacklelike. We now come to the complementary problem in the relation of stress waves and fracture, that is, the generation of stress waves when brittle fractures propagate through solids. Early experimental work on this includes that of Oi (1966) and Tsai and Kolsky (1967).A theoretical discussion of the waves set up in a rod by a brittle tensile fracture propagatiop through it was first given by Miklowitz (1952). Miklowitz was interested in the phenomenon of rods breaking in two places simultaneously when they were subjected to large tensile stresses, and he put forward an explanation of this phenomenon in terms of the reflection of the outgoing compressive pulse at the free ends of the specimen as a pulse of tension and the interference between this reflected tensile pulse with the flexural pulse that is generated by the fracture. Miklowitz was, however, unable to obtain direct experimental confirmation of his theoretical predictions. Oi (1966) observed the compressive pulses generated by the tensile fracture of metal wires, but since he was primarily interested in strain gage response he did not investigate the generation of flexural pulses and in fact his experimental setup was so arranged that it was insensitive to any flexural waves that were propagated. Tsai and Kolsky studied the generation of the fracture pulses produced when fracture takes place under conditions of the Hertzian impact between a steel ball and a large glass plate. They were able to show that the additional surface pulse generated under these conditions was of the form that might be expected. The experimental work here was simple and straightforward, but the interpretation of the experimental results was rather involved. Phillips (1970) carried out careful experiments on the fracture of glass rods when they fractured under conditions of simple tension and showed that the pulse shapes observed could be very accurately predicted on the assumptions that the stress in the unfractured region remains unchanged during the fracture process and that the velocity of fracture propagation is constant throughout the fracture process. [Phillips in fact assumed that it was 0.38 times the velocity of extensional waves in glass rods. This is about
36 1
Experiments in Solid Mechanics
the observed value and has some theoretical backing (Roberts and Wells, 1954).] On these assumptions it is possible to predict both the change in axial force and the bending moment set up during the fracture process. As a result of the change in axial force on the fracture plane a compressional stress pulse is set up, and since the remaining stress is not symmetrical about the diameter of the rod, a bending moment is produced, which results in a flexural pulse being propagated. Figure 27 shows the calculated shapes of the two stress pulses produced
I UO
I I
I I I I
- - - - v-------
B t
FIG.27. Pulse shapes calculated for fracture plane with specimen undergoing tensile fracture.
by the fracture. The way the fracture is assumed to spread across the section of the rod is shown in the inset on the figure. The two stresses c L ( t ) and a,(t) are both plotted as functions of time t from fracture initiation. These pulse shapes are calculated for the plane of fracture (although, of course, in reaiity the stress in the fracture plane is very much more complicated than this), and the shapes are relevant only in that they enable us to predict the observed pulse shapes some distance dowq the rod where the stress distribution has settled down. If at time t the axial force in the rod has changed by an amount F ( t ) as a result of the growing fracture, oL(t)is defined as F(t)/A,, where A, is the cross-sectional area of the rod. Thus oL(t)represents the longitudinal stress produced by the compression pulse some distance down the rod if dispersion
H . Kolsky
362
effects are neglected, whereas oF(t)represents the shape of the stress at the point of initiation of a hypothetical flexural pulse generated at the fracture plane, which has a moment about the neutral axis equal to that produced by the fracture. If the moment produced by the fracture is M ( t ) , then oF(t)is defined by aF(t)= M(t)R/Z,where R is the radius of the rod and I the second moment of the cross section about a diameter. Now the dispersion produced in the longitudinal waves is very small under these conditions, and a strain gage mounted on the surface of the rod will observe a pulse shape close to that shown for aL(t).Flexural waves, however, are inherently dispersive and the flexural pulse will continuously change in shape as it travels down the rod. Phillips used the Timoshenko equations to allow for this dispersion, and Fig. 28 shows a comparison between the observed output of a strain
I
FIG. 28. Comparison between observed (-) diameter glass.rod 6.45 in. from fracture plane.
and calculated ( 0 )strain pulses in ;-in,
gage mounted 12.9 diameters away from the fracture surface in line with the point of initiation of the fracture and the predicted pulse shape. It can be seen that the agreement between theory and experiment is very close. In Phillips’ work it was found that the fracture could be initiated at any prescribed point on the rod by filing a small groove at that point. When a groove was ruled around the circumference of the rod, it was hoped that fracture might start simultaneously at several points on the circumference and grow onto the axis, so that although a compressional pulse would be generated there would be no net moment M(t), and the flexural pulse would be suppressed. It was found to be impossible to do this since fracture always started at one point on the circumference and spread in the same way as when a small surface flaw was introduced. Now Lee and Kolsky (1970) investigated the fracture pulses set up by Hopkinson fractures, where a rod is
Experiments in Solid Mechanics
363
broken as a result of the reflection of a compressional pulse at its free end. In such experiments when a circumferential groove was drawn in the expected place of spalling, it was found that fracture was initiated at a number of points around the circumference simultaneously, and only a compressional pulse was generated. The duration of this compression pulse was found to be one-half that of the pulse produced when the fracture had to traverse the entire cross section (see also Kolsky, 1973). The usual way glass rods are broken is a result of bending rather than pulling them. Now in pure flexure one-half the cross section is subjected to tensile stress while the other half is in compression, and if one follows the assumption that Phillips found to work so well for tensile stress, namely, that the stress in the plane of the fracture does not change during the fracture process, one is faced with a situation where the fracture would be expected to run up to the neutral axis and then stay there. This is clearly contrary to experience, and experiments were carried out to determine what, in fact, did happen under these conditions. It was soon established that the duration of the fracture process was considerably longer for flexure than for tension, and it was found that the duration increased linearly with the length of the rod specimen (Kolsky, 1973; Bodner, 1973). The fracture was, in fact, only completed when the compressional pulse generated by the early part of the fracture was reflected at the free ends of the rod as a pulse of tension. The results of Bodner’s experiments showed that the fracture did penetrate the region initially under compression and more recent high-speed photographs of the fracture process under these conditions (Kinra and Kolsky, 1975) have shown that the fracture crosses about three-quarters of the cross section before it slows down appreciably, and thus it traverses a region that was initially under compression. However, a fracture cannot penetrate a compressive region, and in order that the crack should continue to propagate it is necessary that the stress in front of the crack tip remain tensile. In the same paper it was shown that the pulses emitted by the fracture as seen further down the rod were consistent with the view that the stress remained unchanged in the unfractured region as the crack progressed through the specimen, and it is proposed that this apparent paradox may be resolved if it is assumed that the stress field in the immediate vicinity of the crack has been built up by the superposition of two stress-wave fields, one produced by a plane longitudinal wave, which is emitted from the crack surface, and the other a cylindrically divergent tensile pulse, which originates at the crack tip. The sum of these two waves results in tensile stress being maintained in front of the crack tip, but since the cylindrical wave attenuates rapidly with distance the pulse that is observed some distance from the fracture plane can either increase or decrease in magnitude as the fracture grows.
364
H . Kolsky
VII. Conclusion
In this article a number of experimental techniques in the field of solid mechanics have been described and discussed and the way they have enabled theoretical advances to be made and physical assumptions to be either validated or shown to be erroneous has been outlined. In the history of solid mechanics often large fields have been opened up with very few physical assumptions; for example, the whole theory of linear elasticity is the consequence of a linear relation between the stress tensor and the strain tensor. In other fields, theory has had to rely very much more heavily on experimental observations. For example, the mathematical theory of linear viscoelastic behavior depends on the applicability of Boltzmann’s principle of linear superposition, and the establishment of this principle for real solids has involved very extensive experimental work. Now unfortunately, once one departs from Hookean elasticity there are no other generalizations of so sweeping a character and there has been an unfortunate tendency during the last two or three decades by so-called natural philosophers to substitute elegance for relevance, and generality for reality. Whereas random experimentation generally leads nowhere, and a sound theoretical background is almost essential for the design of meaningful experiments, the development of theories mainly to illustrate the power of the mathematical tools the writer has at his disposal, without any noticeable contact with reality, is equally sterile. There are many good examples of the mutually beneficial intertwining of theory and experiment in the field of solid mechanics and perhaps the most successful ones are described by Zener (1948) in his excellent book on the response of metals to applied forces. In this work he describes the response of metals to various types of deformation and discusses the many and varied microscopic processes that are responsible for the observed behavior. The many triumphs of this approach are described, as well as many situations where the problems are too difficult for theoretical treatment. It is perhaps ironic that the resurgence of the abstract approach to the subject called continuum mechanics owes its inception to the classical work of Rivlin (1947, 1948) on the theory of the deformation of rubber. This work, although mainly mathematical in nature, is intimately related to the behavior of a real material, and the weaving of theory and experiment in the whole of Rivlin’s work on rubber elasticity is an excellent example of what theory is able to accomplish under favorable conditions. Most practitioners in the field would agree with this. Nevertheless a large fraction of the papers that are now published on problems in continuum mechanics appear to be very remote from any possible application to the real world. The time-worn
Experiments in Solid Mechanics
365
cliche that “one never knows what may ultimately turn out to be useful” is by now wearing a little thin, and one would prefer to have the franker statement that the problem interested the writer and looked as if it might have some hope of solution, but that any possible practical application was a matter of little concern to him. To sum up, there is much to be said for the view that experimenters ought to have a better understanding of the underlying theory of their subjects but there is at least as much in the complaint that applied mathematicians should become aware of the existence of a real world and be prepared to leave the field of elegant speculation to pure mathematicians, who are paid (or not paid!) to study it.
ACKNOWLEDGMENT The author wishes to thank Mrs. E. Fonseca for typing the manuscript and Miss E. Addison for the drawings. He also wishes to acknowledge with thanks the support he has received from the Advanced Research Projects Agency Contract SD-86 between the Agency and Brown University. REFERENCES
ABRAMSON,H. N., PLASS,H. J., and RIPPERGER, E. A. (1958). Stress wave propagation in rods and beams. Advan. Appl. Mech. 5, 111-194. ALTER,B. E. K., and CURTIS,C. W.(1956). Effect of strain rate on the propagation of a plastic strain pulse along a lead bar. J. Appl. Phys. 27, 1079-1085. ANDRADE, E. N. DEC.(1914). Flow of metals under large constant stresses. Proc. Roy. SOC.,Ser. A 90, 329-342. BALLOU, J. W.,and SILVERMAN, S. (1944). Young’s modulus of elasticity of fibers and films by sound velocity measurements. J. Acoust. SOC. Amer. 16, 113-1 19. BELL, J. F. (1951). “Propagation of Plastic Waves in Pre-Stressed Bars,” Rep. No. 5. Navy Contract NG-ONR-243 VIII. Johns Hopkins Univ., Baltimore, Maryland. BELL,J. F. (1956). Determination of dynamic plastic strain through the use of diffraction gratings. J. Appl. Phys. 27, 1109-1 113. BELL,J. F. (1964). A generalized large deformation behavior of face cubic solids-high purity copper. Phil. Mag. 10, 107-126. BELL, J. F. (1965). The dynamic plasticity of metals at high strain rates. An experimental generalization. In “Behavior of Materials under Dynamic Loading,” (N. J. Huffington Jr., ed.), pp. 19-41. Amer SOC. Mech. Eng., New York. BELL, J. F. (1973). “The Experimental Foundations of Solid Mechanics,” Vol. VIa/l, “ Mechanics of Solids.” Encyclopedia of Physics (S. Flugge, ed.). Springer-Verlag, Berlin and New York. BENBOW, J. J. (1953). An apparatus for measuring the dynamic shear elasticity and viscosity of solids at high audio frequencies. J. Sci. Instrum. 30,412-413. BENBOW, J. J. (1954). The dynamic mechanical properties of some organic glasses. Proc. Phys. SOC., London 67, 120- 130.
366
H.Kolsky
BODNER,S. R. (1973). Stress waves due to fracture of glass in bending. J. Mech. Phys. Solids 21, 1-8. BOLTZMANN, L. (1876). Pogg. Ann. Phys. Chem. 7, 624. BRIDGMAN, P. W. (1931). “The Physics of High Pressures.” Bell, London. CAMPBELL, J. D. (1973). Dynamic plasticity: macroscopic and microscopic aspects. Mater. Sci. Ew. 121, 3-21. CHRISTIE,D. G. (1952). An investigation of cracks and stress waves in glass and plastics by high-speed photography. J. SOC.Glass Technol. 36, 74-87. COULOMB, C. A. (1784). Recherches theoriques et exptrimentales sur la force de torsion et sur IBlasticite des fils de metal. Hist. Acad. Roy. Sci. Paris pp. 229-269. DAVIES, R. M., and JAMES, H. (1934).Electrically maintained vibrating reed for determination of Young’s modulus. Phil. Mag. 18, 1023-1052. DAVIS,E. D. H., and HUNTER,S. C. (1963). The dynamic compression testing of solids by the method of the split Hopkinson bar. J. Mech. Phys. Solids 11, 155-179. DONNELL, L. H. (1930).Longitudinal wave propagation and impact. Trans. ASME 52,153-167. DWALL,G. E. (1966).Shock waves in the study of solids. In “Applied Mechanics Surveys.” (N. Abramson, H. Liebowitz, J. Crowley, and S. Juhasz, eds.), pp. 869-883. Spartan Books, Washington, D.C. D u w z , P. E.,and CLARK,D. S. (1947). An experimental study of the propagation of plastic deformation under conditions of longitudinal impacts. Proc. ASTM 47, 502-532. FERRY, J. D., and FITZGERALD, E. R. (1954).Dynamic rheological properties of linear polymers. Proc. Int. Con$ Rheol. 2nd Oxford, 1953. pp. 140-149. Butterworth, London. GOLDSMITH, W. (1960). “Impact.” Arnold, London. HILLIER,K. W. (1949).The measurement of some dynamic constants and its application to the study of high polymers. Proc. Phys. SOC, London, Sect. E 62, 701-713. HILLIER,K. W. (1950). Measurement of the dynamic elasticity of rubber. Trans. Inst. Rubber Ind. 27, 64-77. HILLIER,K. W. (1951). A vibrating cantilever method for the investigation of the dynamic elasticity of high polymers. Proc. Phys. SOC.,London, Sect. E 64, 998-1005. HILLIER,K. W., and KOLSKY, H. (1949). An investigation of the dynamic elastic properties of some high polymers. Proc. Phys. SOC.,London, Sect. E 62, 1 1 1-121. HOPKINSON, B. (1912). The effects of the detonation of gum-cotton. Scientific Papers, p. 461. (Cambridge Univ. Press, 1921, p. 498.) HOPKINSON, J. (1872). On the rupture of an iron wire by a blow. (Scientific Papers, 1905, pp. 316-320.) HUNTER,S. C. (1960). Viscoelastic waves. In “Progress in Solid Mechanics.” (I. M. Sneddon and R. Hill, eds.) Vol. 1, pp. 1-57. North-Holland Publ, Amsterdam. JAHSMAN, W. (1971). Reexamination of the Kolsky technique for measuring material behavior. J. Appl. Mech. 38, 75-82. KINRA, V. K., and KOLSKY,H. (1975). “Interaction of Brittle Fractures and Stress Waves,” Tech. Rep. No. 12, NSF-GK-3798. Brown Univ., Providence, Rhode Island. KOLSKY, H. (1949).An investigation of the mechanical properties of materials at very high rates of loading. Proc. Phys. SOC.,London, Sect. E 62, 676-700. KOLSKY, H. (1953). “Stress Waves in Solids.” Oxford Univ. Press (Clarendon), London and New York. (Dover reprint, 1963.) KOLSKY, H. (1956).The propagation ofstress pulses in viscoelastic solids. Phil. Mag. 1,693-71 1. KOLSKY, H. (1969).The production of tensile shock waves in stretched natural rubber. Nature (London), 224, 1301. KOLSKY,H. (1972). Some recent experimental investigations in stress-wave propagation and fracture. In “Dynamic Response of Structures.” (G. Herrmann and N. Perrone, eds.), pp. 327-343. Pergamon, Oxford.
Experiments in Solid Mechanics
367
KOLSKY,H. (1973). The stress pulses propagated as a result of the rapid growth of brittle fracture. Eng. Fracture Mech. 5, 513-522. KOLSKY,H., and DOUCH,L. S. (1962). Experimental studies in plastic wave propagation. J. Mech. Phys. Solids 11, 195-223. KOLSKY, H., and LEE,S. S. (1962). “The Propagation of Stress Pulses in Linear Viscoelastic Media,”’Tech. Rep. No. 5, Contr. Nonr 562(30).Brown Univ., Providence, Rhode Island. KOLSKY,H., and RADER,D. (1968).Stress waves and fracture. I n “Fracture” (H. Liebowitz, ed.), pp. 553-569. Academic Press, New York. KOLSKY, H., and SHEARMAN, A. C. (1949).Investigation of fractures produced by transient stress waves. Research (London)2, 384-389. LEE,E. H. (1956). Wave propagation in anelastic materials. I n “Deformation and Flow of Solids” (R. Grammell, ed.), pp. 129-136. Springer-Verlag, Berlin and New York. S. T. (1956).Analysis of plastic deformation in a steel cylinder striking a LEE,E. H., and TUPPER rigid target. J. Appl. Mech. 21, 63. LEE,J. P., and KOLSKY, H. (1970). “ Fractures Produced by Stress Pulses,” Tech. Rep. No. 10, NSF-GK-3798. Brown Univ., Providence, Rhode Island. LETHERSICH, W. (1950). The rheological properties of dielectric polymers. J. Appl. Phys. 1, 294-301. LIFSHITZ,J., and KOLSKY, H. (1965). The propagation of spherically divergent stress pulses in linear viscoelastic solids. J. Mech. P h p . Solids 13, 361-376. LINDHOLM, U. S, ed. (1968). “Mechanical Behavior of Materials under Dynamic Loads.” Springer-Verlag, Berlin and New York. MALVERN, L. E. (1951). Plastic wave propagation in a bar of material exhibiting a strain-rate effect. Quart. Appl. Math. 8, 405-411. MALVERN, L. E. (1965).Experimental studies ofstrain-rate effects and plastic wave propagation in annealed aluminum. In “Behavior of Materials under Dynamic Loading” (N. J. Huffington Jr., ed.), pp. 81-92. Amer. SOC.Mech. Eng., New York. MARKOVITZ, H., YAVORSKY, P., HARPER, R. C., ZAPUS,L. J., and DEWITT,T. W. (1952).Instrument for measuring dynamic viscosities and rigidites. Rev. Sci. Instrum. 23, 430-437. MIKLOWITZ, J. (1952). Elastic waves created during tensile fracture-The phenomenon of a second fracture. J. Appl. Mech. 20, 122-136. MOFFETT,M. B. (1971). “Experimental Demonstration of a Viscoelastic Cross-Over Phenomenon. The Reflection of Stress Pulses at the Boundary between Nearly Elastic and Lossy Viscoelastic Media Whose Impedances Are Matched,”Tech. Rep. No. 18, DA/358/17. Brown Univ., Providence, Rhode Island. MOFFETT,M. B. (1973). Experimental demonstration of a viscoelastic cross-over effect. J. Acoust. SOC.Amer. 53, 1749-1751. NOLLE,A. W. (1947). Acoustic determination of the physical constants of the rubber-like materials. J. Acoust. SOC.Amer. 19, 194-208. 01,K. (1966). Transient response of bonded strain gages. Exp. Mech. 6, 463. PARFITT,G. G. (1949). Energy dissipation in solids at sonic and ultrasonic frequencies. Nature (London) 164, 489-490. PHILLIPS, J. W. (1970). Stress pulses produced during the fracture of brittle tensile specimens. I n t . J. Solids Srruct. 6,1403-1412. QUIMBY, S. L. (1925). Experimental determination of viscosity of vibrating solids. Phys. Rev. 25, 558-573. RAKHMATULIN, H. A. (1945). On plastic waves of loading. (In Russ.) Prikl. Mar. Mekh. 9, 91. RAMBERG, W, and IRWIN, L. K. (1957). A pulse method for determining dynamic stress strain relations. Proc. I n t . Congr. Appl. Mech. 9th, Brussels, 1956 8, 480-486. C. (1939). Intercrystalline thermal currents as RANDALL, R. H., ROSE,F. C.,, and ZENNER, sources of internal friction. Phys. Rev. 56, 343-348.
H.Kolsky
368
RINEHART, J. S.,and BARSON, J. (1954). “ Behavior of Metals under Impulsive Loads.” Amer. SOC.Metals, Cleveland, Ohio. (Dover reprint, 1965.) RIPPERGER, E. A., and YEAKLEY, L. M. (1963). Measurement of particle velocities associated with wave propagation in bars. Exp. Mech. 3, 47-56. RIVLIN,R. S. (1947). Torsion of a rubber cylinder. J. Appl. Phys. 18, 444-449. RIVLIN,R. S. (1948). Large elastic deformations of isotropic materials. Phil. Trans. Roy. SOC. London, Ser. A 240,459-525; 241, 379-397. ROBERTS, D. K., and WELLS,A. A. (1954). The velocity of brittle fracture. Engineering (London) 178, 820.
STERNGLASS,E. J., and STUART,D. A. (1953). An experimental study of the propagation of transient longitudinal deformation in elasto-plastic media. J. Appl. Mech. 20, 427-434. TAYLOR, G. I. (1946). The testing of materials at high rates of loading. (James Forrest Lecture.) J . Inst. Civil Eng. 26, 486. TSAI,Y. M., and KOLSKY, H.(1967). A study of the fractures produced in glass blocks by impact. J . Mech. Phys. Solids 15, 263-278. VON KARMAN, T., and Duwez, P. E. (1950). On the propagation of plastic deformation in solids. J . Appl. Phys. 21, 981. H. (1935). Internal dissipation of solids for small cyclic strains. WEGEL,R. L., and WALTHER, Physics (New York) 6, 141. YOFFE,E. H. (1951). The moving Griffith crack. Phil. Mag. 42, 739-750. ZENER,C. (1948). “Elasticity and Anelasticity of Metals.” Univ. of Chicago Press, Chicago, Illinois.
Instability of Surface and Internal Waves CHIA-SHUN YIH Department of Applied Mechanics and Engineering Science University of Michigan, Ann Arbor, Michigan
I. General Introduction
. .. .. . . .. . . .. .. .. .. . . . .. .. .. .. .. .. ..
I. General . . . .. . . .. . . .. .. . . .. . . 11. InstabilityIntroduction of Surface Waves 11. A. Instability of Surface Introduction . . , Waves . . . . . . . . . ... . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A. The Introduction . . . . . .. . . . . . . . . . . . . B. Primary Flow ~
~
C. Formulation of the Stability Problem . . ., . . . . . D. Demonstration of Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Demonstration of Instability . . . . . . . . . . . . . . . . . . . .
G. Effects of Surface Tension . G. Effects of Surface Tension
..............................
I. Conclusions J. Discussion.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . J. Discussion.. . . . . . . . . . . . . . . . . . .. . . . . . 111. Instability of Internal Waves .. . . . . . . . A. Introduction ................ B. Stationary Inter . . . . . . . .. . . . . . . . . . . . .
D. Stationary Waves in a Continuously Stratified Fluid F. Resonance across Modes
.. .. .. .. . . . . .. .
..............................
. . . . , . . . . . . . . . . . , , . .. . . . . . . . . ,
.
I. Remarks. .. .. .. .. .. .. . . . . . . . . . . ........... .. .. .. .. . . . . . . . . . . .. .. ....... .. . . . . . . References . . . . .. ......... .. .. .. .. .. .. ............. .. .. .. .. .. .. .... . . . . . . . . , . , . . . . Note Added in Proof
369 3691 37 37 11 37 37 1 372 372 375 375 377 377 382 382 384 384 385 385 386 3861 39 39 11 39 39 1 393 393 393 393 394 394 396 396 408 408 410 410 417 417 418 418 418 418 418 418 419 419 419 419
I. General Introduction The instability of waves arising from the interaction of wave trains of different wave numbers and frequencies has been a subject of much recent study. The instability of surface waves propagating in water of uniform 369
370
Chia-Shun Yih
depth, finite or infinite, has been investigated by Whitham (1966) by a variational method and by Benjamin (1967) by considering two side-band wave numbers and frequencies in addition to the wave number and frequency of the primary wave train. Hasselmann (1967) demonstrated that, for dispersive waves, if the wave numbers k,, k,, and k , and their corresponding frequencies o,, o,, and o3 satisfy the conditions
k3 = k ,
+ k2
and
o3 = o1 + 0 2 ,
then there is interaction between the three wave trains, and if the wave train with k l is of finite amplitude and the other two wave trains are of infinitesimal amplitudes, then the finite wave train is unstable in that its energy will be transferred to the other two wave trains. But Hasselmann assumed the dispersion relation between o and k to be given by the linear theory, although of course the interaction between wave trains is a result of nonlinearity. Thus it is not possible (Phillips, 1966) for surface waves to satisfy Hasselmann’s conditions. Although Davis and Acrivos (1967) did find internal-wave triads satisfying Hasselmann’s conditions, they did so only by resorting to different modes of the internal waves. When these conditions are satisfied, instability does result, as Hasselmann predicted. This article is divided into two parts. The fist part treats the instability of surface waves and the second part the instability of internal waves. In each part the instability of stationary waves due to a flow over a wavy boundary is treated first, and then the theory is applied to progressive waves propagating in water (stratified or not) of constant depth. Whether the waves are stationary or progressive, instability is found whenever Hasselmann’s conditions are met. Yet the results presented herein are, we think, outside the pale of Hasselmann’s theorem, because (a) we consider a flow over a wavy boundary whereas Hasselmann treated free waves, or (b) for freely propagating waves we consider the variation with amplitude of the wave velocity, whereas Hasselmann did not. It is hard to see how all the complicated details in this article that are necessary for reaching our results could be obviated in the proof of Hasselmann’s theorem, and yet these results could be directly obtained by an application of that theorem. When the present theory is applied to progressive surface or internal waves, instability is found only if the wave velocity increases with the amplitude of the basic wave train. The mechanism of instability is different from that of Whitham (1966) and Benjamin (1967) in that the disturbances considered in this article interact only once with the basic waves to achieve resonance, whereas in Benjamin’s mechanism of instability they interact twice. (Whitham’s results agree with Benjamin’s, so that their mechanisms must be identical or equivalent.) A further difference, not unrelated to the
Instability of Surface and Internal Waves
371
one just mentioned, is that the wave numbers of the disturbances are not necessarily near the wave number of the basic wave train. We shall compare our results with Benjamin’s later and give explanations for the differences.
II. Instability of Surface Waves A. INTRODUCTION This part is primarily a study of the stability of stationary waves that form on the surface of a liquid layer flowing over a wavy bottom, followed by an application to free waves. At first sight the mechanism of instability seemed to have much to do with that of water in a vertically oscillating container (Benjamin and Ursell, 1954),since the liquid stream may be thought of as rising and sinking as it moves over the wavy bed. This conjecture turned out not to be fruitful. The reason is twofold: the conjecture has force only if the wavelength is long, but if the wavelength is long the vertical acceleration is very small, and its slight spatial variation is able to destroy any tendency toward the kind of instability studied by Benjamin and Ursell. The primary cause of instability turns out to be a kind of resonance. In the running stream it is possible for two perturbation wave motions to exist, which have the same frequency but different wave numbers m and m‘ (> m), whose difference is the wave number k of the primary wave motion. These two wave motions interact with the primary wave motion and can thus grow in amplitude as a result of resonance. The major part of this paper is devoted to finding the condition of instability of stationary gravity waves in a liquid flowing over a wavy bottom, and the rate of growth. The effects of surface tension are then investigated. In all cases, when the flow is unstable, the rate of growth of the disturbances is proportional to the amplitude of the primary waves. When the analysis is applied to progressive gravity waves by adopting a moving frame to render them stationary, the Froude number is the ratio of the wave velocity to long-wave velocity, and therefore depends on the wave number. It turns out that progressive gravity waves are unstable for all nonzero wave numbers. This result differs from the result of Benjamin (1967) and Whitham (1966)that progressive gravity waves are stable if the dimensionless wave number is less than 1.363. The theories of Benjamin and Whitham agree, and Benjamin’s, in particular, has been abundantly verified by experiments conducted by Feir (see Benjamin, 1967), so that these theories can be considered well established. The present linear theory does not contradict them; rather it presents a new mechanism of resonance,
372
Chia-Shun Yih
which renders all progressive gravity waves unstable, although for deepwater waves the instabilities found by Benjamin and Whitham are often stronger and always more detectable. Detailed comparison with Benjamin’s results will be given, with explanations for the differences. However, it should be noted here that since the Froude number is no longer arbitrary, the growth rate of disturbances in a progressive wave train can be of the order of 8’ (j3 being the amplitude of the primary waves), as shown by (2.59) in Section 1,H. It can also be of the order of j3.Whether it is of the order of j3 or P2 or of some other order of j3 depends on the wavelength of the waves under study and on the value of j3 itself. As will be seen in Section II,D, the method of approach used here is similar to that used by Davis and Acrivos (1967) to study the stability of progressive internal waves. Their method, like the present one, also gives a growth rate proportional to the amplitude of the waves when they are unstable, in contrast with the results of Benjamin, as a result of the difference in the mechanism of resonance. The difference between the present work and the work of Davis and Acrivos is twofold. First, stationary waves over a wavy boundary are considered, with the Froude number arbitrary. This is unlike the progressive waves, which when made stationary correspond to a definite Froude number. Second, when the stability of progressive waves is considered, instability is found in this paper by considering the increase of the velocity of progressive waves with amplitude, whereas Davis and Acrivos did not consider this increase [or decrease, since they considered internal waves, for which the wave velocity may increase or decrease with amplitude, as shown by Yih (1974)], and found instability only by considering the interaction of various internal modes. B. THEPRIMARY FLOW The primary flow is a steady irrotational flow over a wavy bottom and with a free surface where the pressure is constant. Since it is the flow at the free surface that is the most important, the shape of the bottom being of only secondary importance, we shall use, for the primary flow, a flow that satisfies the nonlinear free-surface condition exactly, with a wavy streamline at the bottom. For this purpose we use one of the two formulas given by Richardson (1920). [These two formulas are reducible to each other, as shown by Yih (1957).] The formula we use is
Instability of Surface and Internal Waves
373
in which H ( w ) and the radical are real for real w,
z = x + iy
w = q5 + i$,
and
where x and y are Cartesian coordinates measured in units of depth d, and q5 and $ are, respectively, the velocity potential and the stream function of the primary flow, measured in units of Ud, with U denoting a mean velocity of the flow. The quantity a in (2.1) is defined by a = dF2, F 2 = U2/gd, (2.2) where g is the gravitational acceleration, acting in the direction of decreasing y . Thus (2.1) is in dimensionless terms. The Bernoulli equation for the free surface is, if all quantities are made dimensionlessand q denotes y on the free surface,
42+ a-'q = const,
(293)
where q2 = u2
+'U
I
= dw/dz
12,
with velocity components u and v in the directions of increasing x and y, respectively. If we take $ = 0 on the free surface, and make sure that both H and the radical in (2.1) are real on the free surface (on which w is real), then for the free surface, (2.1) gives
I dwldz l2 = H(w),
(2.4)
and, by integrating the imaginary part of (2.1), q = -aH(w)
+ const.
(2.5)
It is evident then that (2.3) is exactly satisfied. For our purpose, we shall take U ( W= ) 1 - b cos kw,
(2.6)
where
f l < 1. The free surface and the wavy bed are shown schematicallyin Fig. 1, and the disturbed free surface is shown in Fig. 2. It is immediately clear that if /I= 0, Eqs. (2.1) and (2.6)give a uniform flow in the x direction, with dimensionless velocity 1 or dimensional velocity U,which has been used as the velocity scale. For small B, (2.1) and (2.5) give a flow with a wavy free surface at $ = 0 and a wavy bottom at $ = - 1. From (2.5) one sees that the amplitude of the free surface is a/3 and its wave number is k, which is also the wave number of the wavy bed. Were /? zero, d would be the depth of the flow
374
Chia-Shun Y i h
FIG.1. Free-surface flow over a wavy bed.
everywhere. Since B is not zero, we can, for instance, identify d as the depth at a particular section (such as a section at a crest of the wavy bed), and U as the mean velocity at that section. It is important that the region of flow between and
$=O
$=-1
be free from singularities. If B is sufficiently small this can always be achieved, as one can see from (2.1) and (2.4), which never give dz/dw = 0
in the region of flow, for sufficiently small B. From (2.1) we obtain dw/dz = u - iu = H [ ( H - ’
- a2H’2)’’2+ iaH’],
(2.7)
in which H is written for H ( w ) for brevity. For the H given by (2.5), (2.4) becomes 42
= 1 - /3 cos kw = 1 -
cos kf$
(2.8) on the free surface. Another quantity needed later is u-’q2 on the free surface. This can be evaluated from (2.7) and (2.8), and after some straightforward calculation and much simplification, is given by u-’q2 = (H-’- a2H”)”2 = 1
- &(l - 4a2k2)f12 - fj?
cos k+
+ A(7 - 4a2k2)B2 cos 2k4 + 0 ( p 3 ) . (2.9)
FIG.2. Disturbed free surface of a liquid flowing over a wavy bed. The undisturbed free surface is not shown.
Instability of Surface and Internal Waves
375
Higher-order terms can be calculated, but we shall need only terms of the zeroth and first order in 8, so that we shall use u-'q2 = 1 - f / 3 cos k4.
(2.9a)
Series (2.9) is absolutely convergent if 1 ~-
1-8
1 + a2k2f12< 1,
(2.10)
as can be seen from the square-root term in (2.9). Condition (2.10) can always be satisfied with a sufficiently small 8. Finally we note that, with subscripts indicating differentiations, the kinematic condition at the free surface is uqx = v = f#ly.
(2.11)
The flow is shown schematically in Fig. 1. We emphasize that (2.1) has been used only for the convenience of constructing an exact solution for stationary waves. For any other wavy bed and any other solution, there will be a term containing cos k$ in q2 and u-'q2-quantities appearing as coefficients in the crucial free-surface condition (2.21) to be presented laterand the analysis will follow exactly the same line to reach exactly the same results. OF THE STABILITY PROBLEM C. FORMULATION
For the disturbance we assume a complex potential
w' = 4' + iv, where 4' is the velocity potential and $' the stream function of the irrotational flow caused by the disturbance, both regarded as functions of 4 and $. It is well known that since w and w' are analytic functions of z, w' is an analytic function of w,except at the singularities, which are excluded from the fluid domain. Using 4 and $ as independent variables then, we enjoy the advantage of applying the boundary condition at $ = - 1 and the freesurface condition at $ = 0. If s denotes the speed of the combined flow, sz=l
+
d(w dz w')
[=I
d(w + w') dw - ( I = q ' J l + & Jdw' , dz dw
so that on the free surface, with Idw'/dw 1' neglected, (2.12)
376
Chia-Shun Yih
The Bernoulli equation at the free surface, where the pressure is zero, is
4; + 4s’
t F-2(q
+ 11’) = const,
(2.13)
where is the free-surface displacement in the vertical direction, due to the disturbance. Taking the differencebetween (2.13) and (2.3),and using (2.12), we have (2.14)
where we have assigned the constant in (2.13) the value zero because we expect # and q’ to be sinusoidal in 4. If u’ and d are the perturbations to u and u, corresponding to the potential &‘, the kinematic condition on the free surface is,
+
(‘I ‘I’)= 0
+ u’,
(2.15)
where the time t is measured in units of U- ‘d. The difference between this equation and (2.11) is
(2.16) However,
since
u=&x=*y,
u=&=
-I+bx.
As in (2.17) U‘IX
= q2‘14
(2.18)
9
so that the right-hand side of (2.16) is #y
- #:u- Iq2q+ = u#* = u- 1
+ u#* (u + uq2q4)#;.
U-1q2q4(u#@
2
since, by (2.18), q2‘Io = u‘Ix = u.
- u#*)
= u- ‘q2&;. ,
(2.19)
Instability of Surface and Internal Waves
377
Thus (2.16) can be written
(2.20) Combining (2.14) and (2.20), we have
This is the free-surface condition. As to the condition at the solid boundary constituting the wavy bed, it is simply
@*=O
(2.22)
at $ = - 1 .
The velocity potential satisfies the Laplace equation
4;s + 4;*
(2.23)
= 0.
Equations (2.21)-(2.23) are the differential system governing the stability of the disturbance or of the flow.
D. DEMONSTRATION OF INSTABILITY Before presenting the formal demonstration of instability, we shall explain in intuitive terms the motivation for concentrating on disturbances of certain wave numbers. If /3 were zero and a wavy disturbance of wave number y were imposed on the uniformly flowing stream, the solution of (2.23)satisfying (2.22) would simply be
# = C exp i(yq5 - c o t ) cosh y($
+ l),
and (2.21) would give the secular equation (oo - y)2 = F-’y
tanh y.
(2.24)
We shall denote by R and L the right- and left-hand sides of (2.24),respectively. In Fig. 3, R and L are plotted against y, and where the two curves intersect the wave numbers are denoted by m and m‘. We claim that if
m’ - m = k,
(2.25)
the waviness of the bed may cause instability by a sort of resonance. That this is indeed the case will be shown in this section. Since the Lcurve is a parabola touching the y-axis at y = co, and the R-curve rises monotonically as y increases, it is evident that both m and
378
Chia-Shun Yih
Y FIG.3. The existence of wave numbers of resonating modes.
m‘ - m increases with ( i o , from zero onwards. The variations of m and oo with k for various values of F 2 will be given in the next section. Knowing that m and m’ are the crucial wave numbers, and anticipating the repeated interactions of the m-waves and m’-waves with the primary waves with wave number k, we assume a perturbation velocity potential & to have the form
4‘ = e-iut[acosh m(@ + l)ei“
+ b cosh m’(@+ l)eim‘$ m + n = cnPcosh{(m - nk)($ + 1)) exp i(m - nk)+ 1
+
m
dn/3” cosh{(m’ n= 1
+ nk)(@ + 1))exp i(m’ + nk)4],
(2.26)
in which m and m’ satisfy (2.25).Note that the expansion (2.26) implies that the harmonics with wave numbers m - nk and m’ nk are results of interactions of the harmonics with wave numbers m and m’ (which we shall, in anticipation, call the resonant harmonics) with the primary waves with wave number k. Thus the terms under the summation signs in (2.26) are, so to speak, the “entourage of the resonant harmonics. The question may immediately be raised: Why must harmonics of wave numbers m - nk and m‘ + nk be merely the results of interactions of the resonant harmonics with the primary waves? Why can they not exist on their own? The answer to this question is simply that of course they can exist on their own, but when they do their circular frequency (i will not be the same as that for the resonant harmonics, since their (i would be determined from (2.24) with y equal to m - nk or m’ + nk, whatever positive integer n is.
+
”
Instability of Surface and Internal Waves
379
(Indeed, y does not even have to assume these values, but in that case the harmonic with wave number y will not, on its own, have the same u as the resonant harmonics, and the harmonics with wave numbers y and y L- nk cannot belong to the “entourage ” of the resonant harmonics either.) For this reason the harmonics with wave numbers m - nk and m‘ + nk arise only from interactions of the resonant harmonics with the primary waves, and are calculated from nonhomogeneous equations, the stability question being exclusively and once and for all settled by the calculation for the resonant pair alone.? This is an extremely important point for the understanding of what follows. From the following analysis one can see, even if the terms involving wave numbers m - nk and m‘ nk are shown in (2.27) only for n = 1 and not shown in (2.28) at all, how the coefficients c, and d, in (2.26)are determined by (2.21), step by step. Indeed, it can be shown not only for the flow being considered here but also for any flow with stationary waves that expansion (2.26),with c, and d, determined by (2.21),is convergent up to the value of /3 for which the primary flow exists. There is then nothing arbitrary in the form of & given by (2.26) and nothing indefinite or obscure in using (2.21) and (2.26) to determine u and to make conclusions on the stability or instability of the primary flow. As we have said, in this determination the calculation need be done for the resonant pair only. [We note here also that in the application of the theory for stationary waves to progressive waves made stationary, F 2 will depend on /3 and have a term of 0(B2). Inclusion of this amplitude-dependent term in F2,using (2.24) to determine, Once and fur all, rn and m’ for a given k, and then using (2.26)for the analysis, as in the following analysis for stationary waves, does not involve any unallowable truncation and should not lead to any confusion concerning the collection of terms of various orders in 8.1 The first term in (2.21), after a straightforward calculation is then
+
+ terms containing c, and d,, t Even if for higher-order terms in u feedbacks from overtones are needed.
(2.27)
Chia-Shun Yih
380 in which
+ @bm’ cosh m‘, B = b(c - m‘) cosh m‘ + @am cosh m,
A = a(o - m) cosh m
a, = am cosh m,
b, = bm’ cosh m‘.
By virtue of (2.9a), the second term in (2.21) is
F- ’u- ‘q2c#$ = e- iufF-’{(am sinh m - @bm’ sinh m’)ei‘@
+ (bm’ sinh m‘ - @am sinh m)eimU} + terms of O(B’) or involving c, and d, .
(2.28)
Equating terms of the same wave number (m,m‘, etc.) in (2.21), and retaining terms of O(1) and O(B) only for the first approximation, we obtain then a(. - m)’ cosh m
+ ifibm‘(2o - m - m’) cosh m’
= F-’(am sinh m
- &%m’ sinh m’),
(2.29)
and
b(o - m’)’ cosh m’ + fBam(2o - m = F-’(bm‘
- m’) cosh m
sinh m‘ - #am sinh m).
(2.30)
Let (o - m)’ = (oo - m)’
+ A, + O(B’),
(2.31)
so that o - m = (oo- m)[l
+ &(o0
- m)-’
+ O(B’)],
(2.32)
in which A, is of order O(8). Then (2.29) and (2.30) become, upon neglect of terms of 0(p2), d, cosh m
bA, cosh m‘-
Bb [2m‘(200 - m - m‘) cosh m’ + F-’m‘ +4
oo - m’
sinh m’] = 0, (2.33)
+ Ba [2m(200 - m - m’)cosh m + F-’m
sinh m] = 0. go-m 4 (2.34) It is evident from Fig. 3 that oo - m‘ is negative and oo - m is positive, so that their ratio in the first term of (2.34) is always negative. The bracketed terms in (2.33) and (2.34) are both negative. This can be shown in the following way. We know that
(oo - m)’ = F-’m (go
- m’)’
= F-’m’
tanh m
(2.35)
tanh m‘.
(2.36)
Instability of Surface and Internal Waves
38 1
Writing (2.35) as
-
(o0 m)’
+ F-‘(a0 - m) tanh m - F-’a0
tanh m = 0
we have no - m = $[ -F-’
tanh m
+ (F-4tanh’ m + 4F-’a0
the positive sign before the radical being taken because cr0 Similarly, from (2.36) we obtain o0 - m’ = $ [ - F 2 tanh m’
- (F4 tanh’
tanh m)l/’], (2.37)
-m
is positive.
m’ + 4F-’ao tanh n ~ ’ ) ~ ’ ~ ] , (2.38)
the negative sign before the radical being taken because go - m’ is negative. Thus 20, - m - m’ = f[ -F-’(tanh
m
+ tanh m’)-MI,
(2.39)
where
M
=f(m’)-f(m),
f(m) = [F-’ tanh m(F-’ tanh m
+ 400)]’/’. (2.40)
Thus M > 0.
Substituting (2.40) into (2.33) and (2.34), we have
a l l cosh m =
4
(F-’ tanh m
as
go - m’ bAl cosh m’- (F-’ tanh in‘ oO-m 4
+ M)m’ cosh m‘
(2.33a)
+ M ) m cosh m.
(2.34a)
Multiplying (2.33a) by (2.34a), we have
A+--.--
c0 - m B’mm’ (F-‘ tanh m -t- M)(F-’ tanh m’ o0 - m’ 16
+ M).
(2.41)
Thus A! is negative and A1 purely imaginary. Returning to (2.32), we see that Q
= o0
+ $A1(ao- m)-’ + O(B’),
A1/2(ao - m) = kilpIA,
(2.42)
2 > 0.
The choice of the positive sign then gives - iar =
exp (- iao +
la p)t,
and the disturbance (2.26) is unstable, for whatever F2, whatever k, and whatever B, provided that m is not zero. (See Section 11,E.)
382
Chia-Shun Yih
It is important to bear in mind that, given /3, F2, and k, rn and 1 are determined uniquely. The m and rn’ for the unstable disturbance may be quite different from k, and thus not in the neighborhood of k. Furthermore, the present theory is linear, and the growth rate I /3 11 is proportional to I fi I rather than /I2.These facts distinguish the present theory from the theories of Benjamin (1967) and Whitham (1966) for progressive gravity waves. We shall treat the stability of progressive gravity waves in Section II,H, to make a direct comparison. We note that the instability of stationary waves is caused by the resonant interaction of the primary waves with the rn-waves and the rn’-waves. The rn-waves propagate with the flowing stream, but the rn’-waves propagate against it. E. WAVELENGTHS AND GROWTHRATESOF UNSTABLE MODES Given m, we can determine go from (2.24) upon substituting m for y. The other root rn‘ of (2.24) is found by numerical computation and then k is found from (2.25).In this way we obtain corresponding values of go,m, and k. In Fig. 4.m is plotted against k for various values of F2,and in Fig. 5 o0 is plotted against k for the same set of values of F2. We see immediately from Figs. 4 and 5 that the curves go through the origin if F 2 2 1, but intersect the k axis at positive values (kc)of k if F 2 c 1. We have worked with positive k. For negative k, rn and rn’ are negative, but the conclusions remain. By virtue of (2.39), or directly from (2.33) and (2.34), Eq. (2.41) giving 1: canbe written ’
I
84 F 2 =I
k
FIG.4. Variation of m with k.
Instability of Surface and Internal Waves
38 3
k
FIG.5. Variation of u,, with k.
where
G=
1 mm' uo - m 2a0 - m - m' + - F-' tanh m 2 4 o0 -m'
(2t~,-m-m'+-F-'tanhm' 1 2
.
1
(2.43a)
The quantity G is plotted against k in Fig. 6 for various values of F2.It is seen that G is positive for F 2 2 1 and for any positive k, whereas, for F 2 c 1,
k
FIG.6 . Variation of G with k .
384
Chia-Shun Yih
k has to be greater than a cut-off value k, for G to be positive. Thus for F 2 2 1 the flow is always unstable for any nonzero k, whereas for F 2 1 the flow is unstable only if k is greater than a cut-off value k, (or, if k is negative,
-=
smaller than - k,). From Fig. 4 we see that m (and hence m')increases with F 2 for any given k. From Fig. 6 we see that G (hence A,) increases with F 2 for any given k. Thus for any given k, the unstable modes have shorter and shorter wavelengths and become more and more unstable as F 2 increases. Furthermore, it is seen from Fig. 6 that G changes but little between F 2 = 4 and F 2 = 100-a fact of some practical usefulness when estimating G at high Froude numbers. Thus the rather short wavelengths of the surface disturbances and their very strong instability when F 2 is high, as observed in pulp conveyed on Fourdrinier wires (or screens), find their explanation here. It is important to note that if F 2 is sufficiently greater than 1, the growth rate 11, 1 is of the order O(p), whereas the growth rate of progressive waves for the instability mechanism of Benjamin (1967) is of the order? O@') only, in the present notation, and is therefore weaker than that for the present mechanism. However, when progressive waves are made stationary by using a moving frame of coordinates, the F 2 is not arbitrary, and the investigationof instability due to the present mechanism will be presented in Section I1,H.
F. EXTENSION TO OTHER W A V Y FLOWS So far we have only considered the primary flow given by (2.1) and (2.6), but Eq. (2.21) governing the stability of the disturbances is quite general. It is both important and useful to see whether the instability that has just been found will exist for other wavy flows. To reach a decision on this question, we need only to see what forms q2 and u-'q2 assume for other flows, and what effect these forms will have, through (2.21), on the stability. For the flow defined by (2.1) and (2.6), q2 is given by (2.8) exactly and u - l q 2 by (2.9a) if terms of order O(8') or higher are neglected, and the stability analysis shows that only terms of orders O(1) and O(p), with the terms of order O(p)being multiples of p cos k4, are needed for q2 and u- lq2. Now there are infinitely many flows over a wavy bottom for which, in dimensionless terms,
u = 1 - @ cos k 4
+ 0(p2),
u = O(p), (2.44) on the free surface, so that the q2 and u- lq2 on the free surface are given by (2.8) and (2.9a), if terms of order 0 ( p 2 )and higher are neglected. In (2.44),p is an amplitude of wave motion, and the form for u can always be achieved t B is proportional to Benjamin's ka. See the two lines after (2.51).
Instability of Surface and Internal Waves
385
since the origin of 4 is arbitrary. Any potential flow for which (2.44) holds on the free surface has wavy streamlines, any one of which below the free surface can be taken as the solid, wavy bed. For such flows, the stability analysis given in Section 11,Dholds exactly. Hence the conclusions reached in that section are not restricted to the flow given by (2.1) and (2.6), but are much more general. For instance, for gravity waves in a liquid of uniform depth d, the x component of the velocity at the free surface is, when the flow is made steady by a set of moving coordinates and velocity components are measured in terms of the wave velocity, ~ = l - + j 3 ~ 0 ~ k 4 + ~b 02 ~ 2 k 4 + b~ ,0 ~ 3 k 4 + * . * , in which b3 = O(P3), etc. If terms of order higher than O(P) are neglected, u has the form given in (2.44),and the stability analysis in Section II,D applies. If, in this flow, we take a (wavy) streamline between the flat bottom and the wavy free surface as a solid lower boundary, the value of F 2 can be made as large as we please by making the mean depth as small as we please. For F 2 greater than 1 then, that flow is unstable for all positive values of the wave number k. We note, incidentally, that the position of the wavy bed determines F2, and is therefore very important in deciding whether or not the flow is stable. One might say that making any internal wavy streamline rigid destabilizes the flow. b2
= O(j3’),
G. EFFECTSOF SURFACE TENSION Surface tension affects the solution for the primary flow as well as the perturbation flow. Crapper (1957) gave an elegant, exact solution for water waves of arbitrary amplitude and for infinite depth of the liquid, with surface tension fully taken into account. But gravity is entirely neglected in his solution. Fortunately, as the analysis in Section II,C shows, only terms of the first order in the amplitude of the waves are important in the determination of stability, and these terms are easily obtainable by the linear theory, even when surface tension is taken into account. Thus, if the wavy bottom is given in dimensionless terms by y = -1
+ a’ cos kx,
and the mean depth (dimensionless)is 1, the solution for stationary waves is, again in dimensionless terms, q’~ = x
- (a cosh ky
+ b sinh ky) sin kx,
(2.45)
Chia-Shun Yih
386 in which
+ Sk2)b = 0, b = a’[cosh k - (F-’k- + Sk) sinh k ] - l ,
( 2 . 41 (2.47)
S = T/pU’d’,
(2.48)
ka - (F-’
where T is the surface tension, d the (dimensional) mean depth, and U the (dimensional) mean velocity. The free-surface displacement is q = b cos kx
and
- ak cos kx + O(a2k2)= 1 - ib cos k 4 + O(j3’), in which 2ak = B. If terms of order O(b2)and higher are neglected, again 4’ = 1 - j? cos k 4 . u=1
We need, however, to modify (2.14) to
(i+
4’
+ F-’q + S[q(qq&)&]= 0
$)cf
at $ = 0,
(2.49)
in which certain terms definitely of order O(b2)have been omitted. Then the demonstration of instability can be given as in Section II,D, with m‘ and m being now the roots of (go
- y)2 = ( F -
+ S y z ) y tanh y,
(2.50)
and with no determined by the condition m’ - m = k. We shall not pursue the details at this time.
H. INSTABILITY OF PROGRESSIVE GRAVITY WAVES IN A LIQUIDOF CONSTANT DEPTH Benjamin (1967) and Whitham (1966) have studied the stability of gravity waves using two different approaches. They agree that when k (kh or K in their notation) is less than 1.363 the waves are stable. Benjamin uses as the disturbance a side-band of wave numbers and frequencies centered around, and therefore in the neighborhood of, the wave number and frequency of the basic waves. When k > 1.363 the rate of growth of the unstable disturbances is of the order O(f12)in the present notation, O(k2a2)in Benjamin’s notation. We shall show that by the present mechanism progressive gravity waves are
Instability of Surface and Internal Waves
387
always unstable, but that the instability is significant only when Benjamin’s theory predicts stability. If we use a frame of reference moving with the waves, and use our dimensionless notation (the velocity scale U now being the wave velocity c), then for Stokes wavest l p ’ X - 2
/? cosh k(y + 1) sin kx + O(/?’), cosh k
(2.51)
where our /3 corresponds to Benjamin’s (1967, p. 65)
-2kBa coth K,
K = k,d
= k,
d = depth, a = wave amplitude,
where kB is used for the dimensional wave number, to distinguish it from the dimensionless k. We use the /? in this way in order to have agreement with (2.8) and (2.44). The dimensional wave velocity c, which is used as the velocity scale, is given in the present notation by (Benjamin, 1967, p. 66), (2.52)
where
Equation (2.52) is valid only if /3’ < k3 if k is small (see Benjamin, 1967, p. 66). Equation (2.21) still governs stability, and we can repeat the analysis of Section I1,D. Thus we have again (2.41) or its equivalent, (2.43),but we need to investigate the values of m and m’.For this purpose we return to (2.35) and (2.36). If we ignore the term of order O(1’) in (2.52), upon taking the proper root as before, these become
- m = (km tanh m/tanh k)l/’, m’ - go = (km’tanh rn‘ltanh k)’]’.
(2.53)
o,,
(2.54)
Taking the sum of (2.53) and (2.54) and using (2.25) we have km tanh m k = ( tanh k
)”’+(
k(k
+ m) tanh(k + m) ‘ I 2 tanhk
It is then obvious that rn = 0,
1.
(2.55)
m’ = k,
t It is important to note that the (p given by (2.51) and the corresponding stream function now serve as the independent variables in (2.21).
+
388
Chia-Shun Yih
and according to (2.41),A, = 0 and there is no instability.? Any instability then must be due to the term fl’f(k) in (2.52).With (2.52) substituted into (2.35) and (2.36), we obtain the equation
[l
+ ;/?2f(k)]k = right-hand side of (2.55).
(2.56)
From (2.55) and (2.56) it follows that
Thus, for small k, m = *fl’kf(k),
m‘ = k + m.
If terms of order O(fl’) are not neglected, the left-hand sides of (2.53) and (2.54) should be multiplied by
1 + -tPf(k), and from the resulting equations one finds that
m‘ - uo = k
kfl’ + m - __ + O(fl”). 8
For small k, we have, with higher-order terms (in fl) neglected,
F-’ tanh m’ = k,
M = k,
no- m = m
+ O(P4),
so that
A:
= -im2k2f12.
and the growth rate is
- 41 = t k I f l I .
I ~ l P ( ~ 0
(2.57)
Recalling that our fl is Benjamin’s - 2ka coth K and our k Benjamin’s K , we see that the right-hand side of (2.57)is Benjamin’s ka, where k is the dimensional wave number and a the dimensional wave amplitude.
7 Professor 0.M. Phillips kindly showed the author the manuscript of the revised edition of his book (1966).In it he mentions the nonexistence ofresonant wave triads, in agreement with our statement here.
Instability of Surface and Internal Waves
389
For large k (deep-water waves), if we assume k fixed and let B be as small as we please, m will be small. In that case from (2.56) we have
m = lkl/ZB2, 8 m‘ = k + m, go - m = k’l2m = ‘k /3 2, m‘-oo=k+O(B2), F-’=k,
M=k,
so that (2.41) becomes
Jt = -4k2.5m2f12, and the growth rate is
1 R1/2(00 - m ) 1 = ko-’5/3/4fi. We have used c/d as the scale for g o and hence the growth rate also. When d is large this scale is inadequate. If we change the scale for the growth rate to c/L, where L is the wavelength, for large k we have (2.58) Since m is supposed small, /3 4 k - 0 . 2 5 ,hence the growth rate based on c/Lis then very weak for very large k, being of the order k-’/’. If, for large k we assume 1 4 m 4 k, from (2.56) we have m = &kp4, go
- m = (km)’”
= ik/3’,
m‘ = k + m,
m‘ - go = k + O(p2),
F-’ = k = M ,
so that (2.41) becomes 2: = - (&k2/?4)2,
and 12, /2(a0 - m) I = &kfi2.
If again c/L is used as the scale of the growth rate rather than c/d, we have growth rate = in/?’.
(2.59)
Equations (2.57)-(2.59) give the rate of growth of unstable disturbances (the rn-waves and the m’-waves) for long or short waves. But since, whatever the wavelength of the primary waves, m is positive and G is positive in (2.43a), progressive gravity waves of all wavelengths are unstable. The result that long waves are unstable, so long as the wavelength is finite, can be compared with the result of Benjamin (1967) and Whitham (1966) that progressive gravity waves are stable for k < 1.363. The explanation for the difference in results lies in the difference in the mechanism of resonance, theirs being the resonance of disturbances with side-band frequencies with
390
Chia-Shun Yih
the second harmonic of the primary waves, ours being the resonance of two wave trains with the basic harmonic of the primary waves. Benjamin’s theory as applied to deep-water waves has been abundantly confirmed by Feir’s experiments and by observations at the National Physical Laboratory (NPL), reported in Benjamin‘s paper (1967). It is therefore necessary to compare the present results for deep water with Benjamin’s and with available experimental data. We shall first compare the exponential growth rates given by (2.58) and (2.59) with the exponential growth rate of the side-band disturbances given by Benjamin, which is, for deep water,
+kia2w = ip’o,
(2.60) where o is the (dimensional) circular frequency of the primary waves. Since
c/d = o / k g d = w/k, the dimensional growth rate corresponding to (2.58) is z
1 /3 10/4$
k1.25,
(2.61)
and that according to (2.59) is a/3’o/8 k.
(2.62)
In the NPL observations k = 50~17.2,1 1 = 0.34, so that both (2.61) and (2.62) are less than (2.60), but (2.61) is relevant because m is small compared with 1. Hence Benjamin’s mechanism dominates the mechanism treated here, under the experimental conditions at NPL. For the Cambridge experiments o=~R/s,
so that
kB = gn’ = 7/ft. The depth of water was not given. Assuming that it is 3 ft, we have k = 21, and in the range of /Icovered by the experiments, m is small and (2.61) should be used. In this case (2.61)is greater than (2.60)for the smaller values I , but the orders of the of p and less than (2.60) for the larger values / magnitudes are quite the same. One must nevertheless pose the question of why the growth rate (2.61)was not observed, especially for very small /3. The answer is that the dimensionless number 6 representing the fractional deviation from o in the side-band frequencies (1 5 6)w was fixed at 0.1 in the Cambridge experiments, but is far from that for the m-waves and m‘-waves treated here. Thus naturally they cannot be observed.
Instability of Surface and Internal Waves
391
Now the m-waves travel with the stream if the primary waves are stationary, and hence against the primary progressive waves in the fixed frame of reference. The opposite is true for the m‘-waves. Since m is small or at least much smaller than k, the 6 in (1 - 6)ois slightly greater than 1, in any case very near 1. Since m’ is very near k, the 6 in (1 + 6)w is very near zero. Thus the Cambridge experiments, with 6 fixed at 0.1, could not have revealed the m-waves or m‘-waves. Since the frequency of the m-waves is small and the deviation of the frequency of the m’-waves from o of the primary waves is small, to detect these small quantities time records such as shown on p. 63 of Benjamin’s paper must be very long. Analysis with records covering a short period of time can only reveal Benjamin’s instability and would necessarily fail to reveal the instability treated here, even when it happens to be stronger than Benjamin’s (which it is not if the amplitude is not too small). The main conclusion of this section is that all progressive gravity waves are unstable. (See note added in proof on p. 419.)
I. CONCLUSIONS (i) Stationary gravity waves in water (or any liquid) flowing over a wavy bottom are unstable for any nonzero wave number of these waves if F 2 2 1, and for any wave number greater than a critical value k, (depending on F 2 ) if F 2 < 1. (ii) Gravity waves of all nonzero wavelengths are unstable. J. DISCUSSION
As mentioned in the introduction, Hasselmann (1967) gave the theorem that for dispersive waves propagating in one dimension, for which the frequency 0 is a function of the wave number k, if there are three wave numbers satisfying the relation k3
= kl
+ k2,
with their corresponding frequencies satisfying 03
= 01
+ 02,
then, on the understanding that the k,-waves are finite and the other two wave trains infinitesimal, one has bl = 0 and a2 = iho2alaf,
(2.63)
a3 = -iho3a:a2,
(2.64)
392
Chiu-Shun Yih
where u l , uz ,and u3 are the amplitudes, however defined, of the three wave trains, the dot indicates the time derivative, the asterisk denotes the complex conjugate, i is, as usual, the square root of - 1, and h is real. From (2.63) and (2.64) it is easy to show that
so that the k2-waves are unstable. The same is true of the k,-waves. In this paper the k, m,and m' correspond to k,, kz , and k , , respectively, and since the frame of reference is stationary with respect to the k-waves, the quantities corresponding to ol,uz , and 0, are, respectively, k,
go
- m, and
go
- m'.
With this in mind, and remembering that u and b in (2.26), (2.33a), and (2.34a) correspond to u2 and u3 in (2.63) and (2.64), we can identify [see (2.4211
with the u2 and i, in (2.63) and (2.64). Upon multiplying (2.33a) by go - m and (2.34a) by go - m', using (2.35) and (2.36), and multiplying b by an appropriate factor, (2.33a) and (2.34a) can be reduced to (2.63) and (2.64). Thus there seems to be a similarity between the present results and the result of Hasselmann embodied in (2.63) and (2.64). Yet the similarity is only superficial. First of all the fact that the bracketed terms in (2.33) and (2.34) are both negative, a fact crucial for the proof of instability and ensuring the existence of a real h in (2.63) and (2.64), has been established in such a special and unobvious way that one can justifiably doubt that a general formula established without regard to the particularity of the problems can apply to the problems considered here u priori. It may be useful to point out further the differences of the present results and the result of Hasselmann. As far as we can see, Hasselmann treated the stability of waves freely propagating in water of uniform depth (i.e., where his theory touches upon the present results). If these are made stationary, the general velocity of flow is not arbitrary but equal to the speed of propagation in quiet water. In this paper, where stationary waves are treated, the general velocity of flow embodied in the Froude number is entirely arbitrary. Thus the conditions on which Hasselmann's theory stands do not seem to be met here, and consequently the instability described here cannot be included in that theorem. Furthermore, whether the waves are due to the wavy bottom or are free
Instability of Surface and Internal Waves
393
waves, the finiteness of their amplitude demands application of the freesurface condition on the free surface and not at a constant elevation. This is taken care of in this article by using d, and II/ as independent variables instead of x and y . There is nothing in Hasselmann’s paper that indicates how this situation is dealt with.
111. Instability of Internal Waves
A. INTRODUCTION Davis and Acrivos (1967) found that Hasselmann’s conditions (mentioned in our Section I) can be satisfied for internal waves in a stratified fluid by resorting to interaction between trains of internal waves not only of different wavelengths but also of different modes. At least one of the wave trains must be of a different mode from the other two. The discovery of Davis and Acrivos is a very interesting one, for the instability they found is truly characteristic of internal waves. It has no counterpart for surface waves in a homogeneous fluid. Our investigations in the instability of internal waves supplement the work of Davis and Acrivos. In the following sections y e will show that Hasselmann’s conditions can be met if (i) the primary wave train is due to a flow of a stratilied fluid over a boundary, or if (ii) it is a wave train freely progressing over a flat bottom, provided.the wave velocity increases with the amplitude. When Hasselmann’s conditions are met, the primary wave train is indeed unstable. In category (i) conditions of stability or instability are found for two cases: (1) waves in two superposed fluid layers, with the upper fluid infinite in extent, and (2) waves in a continuously stratified fluid. The growth rate of the disturbances is given whenever they are unstable. In Case 2 Boussinesq’s approximation is used. We emphasize that the mechanism of instability is, as for surface waves, the resonance of a pair of disturbances with the basic waves produced by flows over wavy surfaces. For interfacial waves the results are nearly the same as for surface waves, but for internal waves in a continuously stratilied fluid the results are somewhat different, the main difference being that these waves are stable for sufficiently high wave numbers and unstable for sufficiently low wave numbers, whatever the internal Froude number may be. The stability of progressive internal waves in an otherwise quiet fluid is briefly discussed after stationary waves over a wavy boundary have been treated.
Chia-Shun Yih
394
B. STATIONARY INTERFACIAL WAVES The primary flow, the stability of which will be studied in the next section, is the flow of two superposed incompressible fluids with a “general velocity U over a wavy bed, which is described by ”
=
-d
+ 7,
cos kx,
(3.1) where x and y are Cartesian coordinates, with y measured in the direction opposite to that of the gravitational acceleration, d and y, are constants, and k is the wave number of the corrugation of the bed. The general velocity U would be the actual velocity of the fluids if y, were zero. The lower fluid has density p, ,and extends from the bed to the interface JJ
Y = 45
q being the (sinusoidal) elevation of the interface above its mean position y = 0. That is to say, if y1 were zero the lower fluid would extend from y = - d to y = 0. The upper fluid has density p1 and extends from the interface to positive infinity. Neglecting viscous effects, we assume the flow in each layer to be irrotational. Velocity potentials 4, and 4, then exist for the upper and lower layers, respectively, both of which satisfy the Laplace equation. Since wave motion must die out as y approaches infinity, we have =
4,
U x + A l e - k y sin kx,
= Ux
(3.2)
+ ( A , cosh k y + B ,
sinh k y ) sin kx.
(3.3) and $, , which are harmonic conju-
The corresponding stream functions gates of q51 and 4,, are
l(li = U y - A , e - k y cos kx, $, = U y + ( A , sinh k y
+ B,
(3.4) cosh k y ) cos kx.
(3.5) We shall give a linear theory for the primary flow, because we need only the first harmonic of the wave motion for our stability study, and a linear theory gives that. At the wavy bed, $, = L U d , so that y = -d
+ U-’(A,
sinh kd
- B,
cosh kd) cos kx.
Uy, = A , sinh kd
- B,
cosh kd.
Hence On the interface,
Instability of Surface and Internal Waves
395
B2 = - A , .
(3.7)
so that
Furthermore, at the interface the Bernoulli equation gives, after the mean quantities have been filtered out, P1
U(41 - w
P2 w
2
x
=
-P - PlSrl,
w,= -P - PZSrl,
-
(3.8) (3.9)
where the subscript x denotes partial differentiation and p denotes the pressure due to waves. The interface displacement '1 is obtained by setting 11/, equal to zero in (3.4), and is q = A, U-' cos kx.
(3.10)
When (3.2), (3.3), and (3.10) are used in (3.8) and (3.9), and p is eliminated, we have (3.11)
A2 = j A l ,
where (3.12) Fi being an internal Froude number. From (3.6) and (3.7), A2 = 7 UYl - A , coth kd, sinh kd
(3.13)
which, together with (3.1 l), gives p z cosh kd
+p,
sinh kd -
kU2
For the convenience of the subsequent development, we shall make the results given above in dimensionless terms. If 4 and 11/ are measured in units of U d and linear dimensions are measured in units of d, after using (3.7), we have
41 = x + a l e - k y sin kx, Ic/l = y cos kx, where
+ (az cosh k y - a , sinh k y ) sin kx, 11/2 = y + (a2 sinh k y - a , cosh k y ) cos ky,
42 = x
(3.15)
Chiu-Shun Yih
396
and k is now and henceforth dimensionless (equal to the original kd). We have a2 = j u l ,
a, = y(cosh k
+j sinh k)-’,
(3.16)
where (3.17) If we write, in accordance with the usage in Section 11,
B = -2a,k, the horizontal dimensionless velocity components u1 and u2 at the interface, to the first order in B, are
so that, to the first order,
u;’q; = 1 - - cos kcp2 .
(3.18)
These formulas will be useful in the next section.
C. INSTABILITY OF STATIONARY INTERFACIAL WAVES It can be readily verified that for waves with the exponential factor exp i(ax - at), where a is the dimensionless wave number and a the dimensionless circular frequency (i.e., with a measured in units of U/d),a is equal to a,, (positive), with a. given by (ao- a)’ = F;’(a
tanh a)(l
+ r tanh a)-’,
with r = p1/p2, (3.19)
provided that y is zero, that is, provided the bottom is flat. Since y is not zero a will differ from a,,, and the purpose of this section is to show that the difference may be a positive imaginary number, signifying instability, if the
Instability o$ Sur$ace and Internal Waves
397
disturbance consists of two wave trains of wave numbers m and m’ satisfying (3.19) and the condition
m’- m = k,
(3.20)
where k is the dimensionless wave number of the primary waves treated in the preceding section. We shall first discuss (3.19)and (3.20)before going on to the study of stability. For a given r and a given Ff, Eq. (3.19)has two roots, one greater than go, which we shall denote by m’,and one less than g o ,which we shall denote by m. That there are two such real roots can be seen in the following way. We shall denote the right-hand side of (3.19) by R and its left-hand side by L. If we plot L against a the curve is a parabola touching the a axis at a = go. If we plot R against a the curve touches the a axis at the origin, rises monotonically (the monotonicity can be easily established) as a increases, but is asymptotically linear for large a. Hence the two curves must intersect at two on its points, one (m)on the descending branch of the L a curve and one (m’) ascending branch. For given k, r, and Ff, we wish to find m and m’ satisfying (3.19)and (3.20). To find m (and therefore m’)and go we adopt the following procedure. Given r and F!, we assign various values to m. Substituting m for a in (3.19),we find go (greater than m). Then with this value of go we solve (3.19) for the other root m‘.The value k is then given by (3.20).In this way we obtain values of m and go corresponding to k, and m-k and uo-k curves can be so constructed. In Figs. 7 and 8 are such curves for various values of FZ and for I = 1. The value 1 is of course never reached by r, and assigning the value 1 to I amounts to adopting the Boussinesq approximation, valid when 1 - r 6 1.
k
FIG.7. Variation of m with k for r = 1 and various values of F f .
398
Chia-Shun Yih
k
FIG. 8. Variation of uo with k for r = 1 and various values of F:.
From Fig. 7 it is seen that for Ff 2 1 we have a nonzero m for a nonzero k, whereas for F: < 1 m is not zero only if k > k,, k , depending on F?.Figures 9 and 10 give m-k and ao-k curves for r = 0.5, and Figs. 11 and 12 give these curves for r = 0.25. It is easy to determine k,. From (3.19) we have F,(a, - m)= J(m),
(3.21)
Fi(o, - m‘) = -J(m’),
(3.22)
k
FIG.9. Variation of rn with k for r = 0.5 and various values of Ff
Instability of Surface and Internal Waves
399
k
FIG.10. Variation of c0 with k for r = 0.5 and various values of F:.
in which m tanh m ‘I2 J(m) = (1 + r tanh m) By virtue of (3.20), the difference between (3.21) and (3.22) is
F ik = J ( m ) + J(m’).
(3. 3
(3.24)
Upon equating m to zero in this equation, we have (3.25)
F i= k - ’ J ( k ) , .I1003616
4
I
3
4
‘/7
5
6
7
8
k
FIG. 11. Variation of m with k for r = 0.25 and various values of Ft.
400
Chia-Shun Yih
k
FIG.12. Variation of o,, with k for r = 0.25 and various values of F ? .
which states that the general velocity U , on which F iis based, is the velocity of internal waves of wave number k in the fluid layers under consideration, if they were at rest and fi = 0. This fact can be readily demonstrated independently in the same way as (3.19).The solution of (3.25) for k is k,. But (3.25) has a real solution only if Fi < 1, for Fi = 1 for k = 0 (very long waves), and the wave velocity decreases with k. We note, however, that the case k = k, must be excluded,? for if (3.25) is satisfied A, is infinite according to (3.14) (where k is still dimensional). This discussion is intended to show that if Fi < 1 but k, < k, Eq. (3.24), in which in' = m k, has a nonzero solution for in. We now give the primary waves a perturbation. The perturbation flow is still irrotational except, of course, at the interface, which is a vortex sheet, as is well known. The velocity potential for the perturbation flow is, for the upper layer,
+
t#~'~
= e-'"'[(o!, exp(
+ irn&l) + p1 exp( -in'$l + iin'&,)],
(3.26)
and, for the lower layer,
t#J'z = e-'"'[a2 cosh
+ l)ei"@*+ fi2 cosh in'($, + l)eim'bz]. (3.27)
It is important to note that at the interface both and $ 2 are equal to zero, is not equal to dZ. but By a procedure similar to that used in Section I1 [see Eqs. (2.12)-(2.14)], the Bernoulli equation at the interface is, after the part corresponding to the ? This exclusion is not necessary if nonlinearity is taken into account.
Instability of Surface and Internal Waves
401
primary flow is filtered out and quadratic terms in the perturbation quantities are neglected, pi Ll C#J'~
+ p' + p1 F2q' = 0
(3.28)
p2 L2#2
+ p' + p 2 F2q' = 0
(3.29)
for the upper layer, and
for the lower layer. In (3.28) and (3.29), p' is the pressure perturbation at the interface, q' the perturbation of q, and
The difference between (3.28) and (3.29) is, after division by p 2 ,
L 2 q 2- rLl#l = - F r 2 q .
(3.31)
As in Section I1 [see Eqs. (2.15)-(2.20)], the kinematic condition at the interface is Ll q' = u; '4: a4% IWi
(3.32)
L2V' = u; l4;
(3.33)
for the upper fluid, and aq21alCI2
for the lower fluid. We recall that, to the first order in u1 = q l
and
fl,
u2 = q 2 .
Applying the operator L2 to (3.31) and using (3.33), we have
(3.34)
since 42
a1842 = 41
a l w l = alas,
s being the distance measured along the interface. The next step is to express
& in terms of 42at the interface. For this purpose we use (3.32) and (3.33), and express q' in the form q' = exp( - iut)[l exp(irn4,)
+ r' exp(irn'~$~)].
(3.35)
402
Chia-Shun Yih
From (3.32) and (3.33) we have, if terms of O(PZ) are neglected, (3.36) which shows that we need not evaluate q’ to the order 41
- 92 = -2ha
cos k 4 2 ,
q2/q1 = 1
fi, since
+ 2hB cos k 4 2 ,
where h = (1 -j)/4J.
(3.37)
Using (3.33), we obtain, upon separating the terms containing exp(im4,) from those containing exp(im’&),
t=
iu,m sinh m + O(B), no- m
r‘ =
iB2mtsinh m’ oo - m’
+ O(B).
(3.38)
Substituting (3.35) and (3.38) into (3.36), we can evaluate a1 and fll in CP; in terms of a2 and fi2, but to do this we have to convert the exponentials exp(im4,) and exp(im’41) into sums of exponentials exp(im4,) and exp(im’4,). With higher-order terms omitted, we have
41- 42= (a, - a2)sin k 4 2 = - 2hB sin k + 2 . k Hence
= cos
m42 + hBm [cos(k - m)& - cos m’&] + O(fi2). k
(3.39)
The term containing cos(k - m)@zcan be omitted, for the components with wave numbers m - nk (n = 1,2,3, . ..) and m’ nk are governed by nonhomogeneous equations with the nonhomogeneous parts produced by the basic components with wave numbers m and m’,which are being considered here. Thus the other components are expressible in terms of the basic components, which determine the stability of the flow. Returning to (3.39), then, we see that, as far as the basic components are concerned,
+
hmB exp(im4,) = exp(im42)- -exp(im’&). k
(3.40)
Instability of Surface and Internal Waves
403
Similarly, exp(im'41) = exp(im'4,)
hm'p
+ 7exp(im&).
(3.41)
We now substitute (3.35) and (3.38) into (3.36), use (3.40) and (3.41), and equate the coefficients of the two basic components exp(im4,) and exp(im'4,). The results are, after a good deal of straightforward calculations, a1 = -az sinh m + hBB2rn' sinh m'
fil
=
).
(3.43)
sinh m ) exp(im'42). + hPaz m2 m'(a - m)
(3.44)
-b2 sinh m' - h/3a2m sinh m
Using (3.40) and 3.41), we have, for
=
-
(
ct2
sinh m
- (pz sinh m'
4; at the interface,
" sinh m' + hPp2 mm(a - m')
This will be used in the first term of (3.34). The next step is to calculate the second term in (3.34). Since it contains the factor q1 - q 2 , which is of O(D), we need only to use the terms of zeroth order in /?in (3.44) for 4; at the interface. A straightforward calculation then gives
[
rL2 (ql - q2)q2
,#I'~= - hpe-iar[p2m'(m- uo)sinh m' exp(im4,) a42
+ a,m(m'
- g o ) sinh m exp(im'4,)],
with all components but the basic ones neglected. The third term in (3.34) is e-'"'FL2
(I
a2m sinh m - B -/lzm'sinh m') exp(im&) 4
P2m' sinh m' - f a 2 m sinh m 4
(3.45)
Chia-Shun Yih
404
Putting (3.44)-(3.46) into (3.34) and equating terms of the same wave number, we have [see Eqs. (2.27), (3.29), and (2.30) for comparison] a2(1
+ r tanh m)(a - m)2 cosh m + $B28(1+ r tanh m’)m’(2a - m - m’) cosh m’ + B2rhbm‘ sinh m’ m’(oo - m) a2m sinh rn - !/3,mr sinh in‘ 4
(3.47)
and
f12( 1 + r tanh m’)(a - m’)’ cosh m‘
+ 4a2/l(1 + I tanh m)m(2a - m - m‘) cosh m + a2rhflm sinh m = F;’
(B2m’sinh m’ - -B4a 2 m sinh m).
(3.48)
Let (cr - m)’ = (ao - m)2
+ Alp + O(B2),
Then cr=ao+
2(00 - m)
+ O(P2),
( 3.49)
and, upon neglect of terms of O(B2),we can write (3.47) as ct,il cosh m
S2 [2(1 + I tanh m’) + 4( 1 +mI ttanh m)
x (2ao - m
- m’) cosh m’ + F ;
+ H1P2 cosh m‘ = 0, where
Hi =
sinh m’] (3.50)
405
Instability of Surface and Internal Waves
Similarly, (3.48) can be written as no- m‘ a. - m
&Al cosh m‘-
ma2 [2( 1 + r tanh m)(200 - m - m’) cosh m + 4( 1 + r tanh m‘)
+ F;’
sinh m] + H 2 a z cosh m = 0,
(3.51)
where
The sign of H 1is determined by the sign of m‘(oo - m ) + 1, m(ao - m’) which is equal to
- .m’J(m) __ + 1 mJ(m‘) by virtue of (3.21) and (3.22). It is a simple matter to show that
“W)
m
so that m’J(m)/mJ(m’)> 1, since m’ > m. Thus the sign of HI is negative. Similarly, the sign of H, is also negative . We now evaluate 2a0 - m - m‘ by the method described in Section I1 [Eqs. (2.35)-(2.39)]. Equation (3.21) can now be written as (o0- m)’
+ F;’(o0
- m)K(m)- F;’a,K(m)
= 0.
(3.52)
where tanh m 1 + r tanh m
(3.53)
ao-rn=i[-F;’K(m)+f(m)],
(3.54)
K(m)= Solving (3.52), we have
406
Chia-Shun Yih
in which
+
f ( m ) = [ F ; 4 K Z ( m ) 4F;2aoK(m)]”z.
(3.55)
The positive sign is taken in (3.54) because a. - m is positive. Similarly, a. - m’ satisfies (3.52), with m replaced by m‘. Hence a.
- m‘ = t[- F;’K(m‘) - f ( m ’ ) ] ,
(3.56)
the negative sign before f ( m ’ ) being taken because oo - m’ is negative. The sum of (3.54) and (3.56) is, after multiplication by 2,
2(2a0 - m - m’) = -F;’[K(m)
+ K(m’)] - M ,
(3.57)
where
M = f ( m ’ ) - f ( m ) > 0.
(3.58)
Returning to (3.50),we see that the term in brackets becomes -D cosh m’, with
D
=~
;
1 + r tanh m‘ 2 [ K ( m ) Mlm’. 1 + r tanh m
+
(3.59)
Similarly, the term in brackets in (3.51) becomes - E cosh m, with E =~
;
1 + r tanh m 2 [K(m’)+ M ] m . 1 + r tanh m’
Therefore we can write (3.50) and (3.51)as a221
=
;(
-
cosh M’
aO-m E a z n 1 = a ( % - H 2 )
7
a2 cosh m
coshm!
9
in which D and E are positive and H I and H, negative. From these two equations we obtain
(3.61)
-=
Since g o - m is positive and a. - m’ is negative, At 0, and the flow is unstable for the disturbances under consideration. Letting
1 = 14 1/2(ao - m),
Instability of Surface and Internal Waves
407
k
FIG.13. Variation of
-A:
with k for r = 1 and various values of FZ.
the growth rate is then AS. In Figs. 13-15, -A: is plotted against k for r = 1, 0.5, and 0.25, and for various values of Ff. We note that both E and H 2contain the factor rn. Therefore so long as rn is not zero the primary flow is unstable. As mentioned before, if F f is not less than unity rn is always positive. If F f is less than unity rn is positive only if k k,, k, being the root of (3.25).
=-
k
FIG.14. Variation of -A: with k for r = 0.5 and various values of F f .
Chia-Shun Yih
408
k
FIG.15. Variation of -1; with k for r = 0.25 and various values of Ff.
In conclusion,stationary internal waves in the fluid system under consideration are unstable for all nonzero wave numbers if F; 2 1, and for all wave numbers greater than k, if F; < 1. We note also that (3.21) and (3.22) show that the m-waves travel with the stream and the m’-waves travel against it.
D.
STATIONARY WAVES IN A CONTINUOUSLY STRATIFIED
FLUID
We shall consider the flow of an incompressible stratified fluid with a general velocity U oover a wavy bottom. If U ois independent of y and p ( y ) denotes the mean density in the absence of wave motion, the stream function is uoy
+ Af(y) cos kx,
where, for the time being, x and y are dimensional, k is the dimensional wave number, andfsatisfies the equation [see Yih, 1965, p. 20, Eq. (31), with U o replacing the c there, or p. 175, Eq. (104), with U oreplacing U and c equal to zero for steady flow]
(pf’y -
(
k2p
+$)f = 0,
(3.62)
with the accent indicating differentiation with respect to y . If the flow is supposed to be confined between two rigid boundaries at distance d apart, the upper one being flat and horizontal and the lower one wavy, we can use d
Instability of Surface and Internal Waves
409
as the length scale. The density on the lower boundary, p o , will be used as the density scale. If we write
then dropping the circumflexes, we have
(pf’)’ - ( k 2 p + F - ’ P ’ ) f = 0,
(3.63)
where the primes indicate differentiationwith respect to the dimensionless y. For simplicity we shall use the Boussinesq approximation, which we expect to yield sufficiently accurate results if the maximum density variation is small in comparison with the mean density. Thus p will be assumed constant and equal to 1 unless it is associated with gravity. If we adopt Boussinesq’s approximation, then whether we assume p to be linear in y or to be an exponential hnction of y the results are the same. We shall then assume p to be a linear function of y, and [note that the present /?, chosen to conform to established usage, differs from that in (3.19)]
p’
-/?.
=
(3.64)
Then (3.63) becomes
f”- (kZ- F r 2 ) f = 0,
(3.65)
where (3.66)
Ff = F2/?.
The condition for f at the upper boundary, where y = 1, is f(1) = 0. The solution of (3.65) satisfying this condition is f(y) = sin ~ (-ly),
I C = ~ FT
- k2.
(3.67)
If the lower boundary is given in dimensionless terms by y = y cos kx,
then since the dimensionless stream function is
h = W ( y ) cos kx, (3.68) $ = Y + h(x, Y ) , and $ can be taken to be zero at the lower boundary, A is related to y by - A sin
Thus, for the primary flow to exist, integer.7
K
K
= y.
is not equal to nn, where n is any
?This condition can be removed by considering nonlinearity.
Chia-Shun Yih
410
We have given the solution for the stationary waves only by a linear theory. As we have stated before, only the first harmonic is of importance, and a nonlinear theory would give substantially the same results. The amp& tude A, being arbitrary (so long as the primary flow exists), can be regarded as the A for the nonlinear theory. The only difference that a nonlinear theory will bring about is the function f ( y ) . But this difference will be of the order O(A),and a glance at (3.68) convinces us that if only terms of the orders O(1) and O ( A )are retained, it is sufficient to use (3.67) and (3.68).
E. INSTABILITY OF STATIONARY WAVESIN A CONTINUOUSLY STRATIFIED FLUID Since the flow is rotational because the fluid is stratified, we must use the Euler equations for the determination of the perturbation flow. If we denote the velocity components of the primary flow by u and u and those of the perturbation by u’ and u’, and write
v=u+ul,
U=u+uI,
the Euler equations are, under the Boussinesq approximation and in dimensionless terms,
au av -at+ u - +axv - = ay Px av av = - p y - F - y p + p’), - + u- + vat ax ay aul
(3.69)
3
a U I
(3.70)
where p is the (total) pressure and p’ the perturbation in density. The equation of incompressibility is
aP’
-
at
a ( p + p’) + v-a / p + p’) = 0, + uax aY
(3.71)
and the equation of continuity for the perturbation flow is aul au’ -+-=o, ax
ay
which permits us to use a stream function $’, in terms of which u’ = 9’Y ’
uI=
-9’
(3.72)
X )
with the subscripts indicating partial differentiation. The velocity components u and u of the primary flow are given by u = $, =
1 + A f ’ ( y ) cos kx,
u=
-i,bx
= A k f ( y ) sin
kx.
(3.73)
Instability of Surface and Internal Waves
411
Elimination of p between (3.69) and (3.70) gives (3.74)
where 1 is the vorticity of the primary flow and flow, so that
-
5 that of the perturbation
5 = -vzI+v.
= -v2*,
The vorticity equation for the primary flow is (3.75)
Taking the difference between (3.74) and (3.75) and neglecting quadratic terms in perturbation quantities, we have L V ~ I ++VE = F - ~ ~ ; ,
(3.76)
where
a
a
at
ax
L=-+--,
E = hy V2$!,- h, V 2 P y+ rl/l V2hx- PxV Z h y.
(3.77)
We now evaluate the right-hand side of (3.76). Using (3.71), we obtain
Lp’ = H,
(3.78)
where
H
= -dpX - v’PY - hy P ’x
+ h p’ x
(3.79)
y ’
In (3.79), the p differs from the p in (3.63) by a term of order O(A),for after the wave motion represented by (3.63), p for the primary flow is a function of both x and y . Indeed,
+ VPy =0
or P J P Y = *XI$, > so that p is a function of I) only. If terms of orders higher than A are neglected, we must have, as the linear theory demands, UP,
P
= 1 - B$,
for otherwise, in the absence of wave motion (i.e., A = 0 and $ = y ) , we should not have (3.64). Thus,
p,
=
-p@x,
p Y = -D$
y
=
- P(1 + hY).
(3.80)
+ Ap; +
(3.8 1)
Let $’ = I+vo + Alpl
+
..a,
p’ = p b
**..
412
Chia-Shun Yih
Then from (3.72), (3.76), and (3.78), we have
L2, v2qo
+ Ff(qo),, = 0,
(3.82)
where L,,, is L when only terms of order O(1) are retained. Let us try the following form for q0:
9o(Y) e x p m - 0 0 t)l. Then
To the order O(l), the boundary conditions are
go(0) = 0 = go(1). The solution of the eigenvalue problem for go is then (a,
- a)'
+
= FT2a2/(a2 n2x2),
(3.83)
where n indicates the mode. Given k, we choose a oosuch that the two roots of a in (3.83), m and m', satisfy
m' - m = k.
(3.84)
As in Section III,C, m is less than ooand m' greater than a, . This can be seen by plotting the two sides of (3.82) against a for any a. and Fi . Thus a.
- m = F;'rn(m2
D 0 - m' =
+ n2x2)-'/',
-F;1m'(m'2
+n n ) 2
2 -112.
(3.85) (3.86)
The difference of the left-hand sides of (3.85) and (3.86) is k. From the right-hand sides we see then that it is impossible to satisfy (3.84) unless
k < 2F;'.
(3.87)
If (3.87) is satisfied, however, it is possible to find m and m' satisfying (3.83) and (3.84), and the quantities m and a. are plotted against k in Figs. 16 and 17 for n = 1, Figs. 18 and 19 for n = 2, and Figs. 20 and 21 for n = 3. These curves are constructed by first assuming m, finding a, by (3.85), and solving (3.83) for m' with the 0, just found. Then k is known, and we have for this k the value for m and the value for ao. All the curves in Figs. 16-21 have vertical asymptotes at k = 2F; We shall then take
$6
= (ael
+ be2) sin nxy,
(3.88)
with el = exp[i(mx - at)],
e, = exp[i(m'x - at)].
(3.89)
Instability of Surface and Internal Waves
413
k
FIG.16. Variation of rn with k for n = 1 and various values of F t .
Note that we use CJ in (3.89) rather than o0, for we anticipate that the terms of order O ( A )in (3.76) and (3.78) will change goby an amount of order O ( A ) . The use of an exponential time factor is certainly justified by the forms of (3.76) and (3.78). Let pb = (a, el
+ b , e 2 )sin nny.
k
FIG. 17. Variation of uo with ,k for n = 1 and various values of F f .
(3.90)
Chiu-Shun Yih
414 E
4
m
?
2
I
0 k
FIG.18. Variation of m with k for n = 2 and various values of Ff.
Then the terms of order 0(1)in (3.78) give a, = pm/(oo- m),
bl
= pm’b/(a0
- m’).
(3.91)
Now, from (3.76) and (3.78), we obtain
L? V2* = -LE
+ F-’H,.
k
FIG. 19. Variation of oo with k for n = 2 and various values of FZ,
(3.92)
415
Instability of Surface and Internal Waves
5
4
m 3
2
I
c
0.5
1.5
I
2
k
FIG.20. Variation of m with k for n = 3 and various values of F:.
Let (0
- m)’ = (oo- m)’ +
A
+ o(A’).
(3.93)
When terms of the zeroth order in A are collected in (3.92),we have (3.82), which is satisfied by (3.88). When collecting terms oforder O(A)in (3.92),we ,I and pb for p’ in E and H except in the term - v’pyin H , because use I& for P
k
FIG.21. Variation of uo with k for n = 3 and various values of F’
Chiu-Shun Yih
416
py contains the term -/?,which is of the zeroth power in A. Furthermore, we use Lo E for L E because E = O(A). With all this in mind, (3.92) becomes
A V2$/'o + L; Vz$/'l
+ Ff(li/;),, = -Lo E + F; ' K ,
(3.94)
where K = /?-'(H - flu'), = [$/'oyhx - $/'oxhy - B-'(pb,h, - &h,)],
.
Straightforward calculation gives
+ Eze,,
LoE = Elel
where E i = E L I+ E i z
Ez = Ezi
9
+ E22
9
with E l l = bA(2m')-'(mfZ + n2n2)oo(oo - m)(oo- 2m')f' sin nny, El, = bAkn7r(2m'2)-1(m'2 + n2a2)oo(oo - m)(oo - 2m')fcos nay, E,, = aA(2m)-'(m2 + ~ ~ a ~ ) -o m')(oo ~ ( o-~2m)f' sin nay,
+
E,, = -aAknn(2m2)-'(mZ n2a2)oo(oo - m')(oo- 2m)fcos nny. As for K, we have
FL2K = K l e l
+ K,e,,
where Kl = K l l
+ K12
9
K, = K , ,
+ K,,
7
with
K l l = bA(2m')-'m(mf2 + n21r2)oo(oo - m')f' sin nay, K , , = bA(2m'2)-1knnm(m'2 + n2d)oo(oo- m')fcos nny,
+ n2a2)oo(oo- m ) f ' sin nay, K,, = -aA(2m2)-'knxm'(m2+ n27r2)oo(oo - m)fcos K Z t = aA(2m)-'m'(m2
nay.
We now multiply (3.94) by 2 sin nlry and integrate from y = 0 to y = 1. [Integrating from the lower boundary? would introduce terms of O(AZ).] The terms involving JI; vanish upon integration by parts. The rest of the t Note that A@\ is of order O ( A )everywhere.It is zero on the lower boundary,at which y is also of order O(A).Hence ,4q1is of order O ( A 2 )and A$',? = O ( A )between y = 0 and the lower boundary. Integration in this interval gives terms of O(A2).
Instability of Surface and lnternai Waves
417
terms, upon separating terms containing el from those containing e2, integrating by parts, and neglecting terms of O(A2),gives -aA(m2
+ n2n2)1,= bA(Zm')-'(m'' + n2n2)aoM1 (3.95)
-bA-
oo - m' (m" oo - m
+ n2n2)1,= aA(2m)-'(m2 + n2n2)ooM2 (3.96)
where 1
I =
jof' sin2 nny dy, + m(oo- m'), - m')(oo- 2m) + m'(a0 - m).
M , = -(oo - m)(oo - 2m') M , = -(go
Now since K is not equal to any integral multiple of n,1 is not zero. Furthermore, M1 and M 2 are equal, as we can see by a glance at their definitions. Hence
- ___ oo oo - m'm 0;(4mrn')-~M;(l -&)(l
1 21 -
+&)12.
(3.97)
Since oo is greater than m but less than m',and since k is less than m', we see that 1;
',
F. RESONANCEACROSS MODES As can be seen from (3.83), (3.85), (3.86), (3.88), and (3.89), Section II1,E treats the resonance of the rn- and m'-waves of the same mode with the basic stationary waves. The restriction that the m- and the m'-waves be of the same mode can be removed. All we need to do is to replace n by n, (an integer) in (3.85) and by n2 (an integer not equal to n,) in (3.86). Then (3.89) and (3.90) become
Po= ael
sin n, ny + be2 sin n2ny,
p' = a l e l sin n l n y
+ b,e,
sin n2ny.
418
Chia-Shun Yih
Subsequent development follows closely the development in Section III,E after (3.90), and again we find the primary waves are unstable whenever nonzero rn and rn’ can be found that satisfy (3.84)and the modified (3.85) and (3.86). Thus resonance across modes is possible and the primary waves can be unstable in many ways.
G. CONCLUSIONS FOR STATIONARY WAVES From the foregoing we conclude that the interfacial waves described in Section II1,B are unstable if the internal Froude number Fi is greater than 1, or if Fi < 1 but the wave number is greater than the k , defined in Section II1,C. Internal waves in a linearly stratified fluid are also unstable, provided the wave number is less than 2/Fi and the Boussinesq approximation can be used. This conclusion must hold also if the fluid is nearly linearly stratified. H. PROGRESSIVE WAVES Progressive waves can be made stationary. The present theory can then be applied to them. When this is done, one finds that the above conclusions hold if the speed of the primary waves increases with the amplitude, and that the waves are stablet otherwise. The speed of progressive internal waves for discontinuous stratifications may increase or decrease with the amplitude (Hunt, 1961). The speed of waves in a continuously stratified fluid decreases with the amplitude if the fluid is weakly and exponentially (or linearly or nearly linearly) stratified, according to Yih (1974). However, without considering amplitude effects on wave speed, Davis and Acrivos (1967) were able to show that instability can result from intermodal interaction.
I. REMARKS As has been noted, the instability of internal waves presented herein, as well as that given by Davis and Acrivos (1967), is a result of resonance when Hasselman’s conditions are satisfied. Again, it is hard to see how Hasselman’s theorem (1967) could be directly applied to reach the conclusion of instability when the demonstration of that theorem is entirely independent of the details of the flow, the boundary conditions, and the exact manner of interaction of the three wave trains. Both the derivation leading to the final results given in this article and that leading to the results of Davis and t Provided that the disturbances are of the same mode.
Instability of Surface and Internal Waves
419
Acrivos are rather involved. It is far from clear that the details could be obviated in the establishment of Hasselman’s theorem and that theorem could still be generally applied, without further ado, to reach the conclusion of instability once the existence of the wave triad satisfying Hasselman’s conditions is assured. ACKNOWLEDGMENTS This work has been jointly supported by the Office of Naval Research and the National Science Foundation. I am grateful to my friends Donald Hall, A. E. Day, and Ting Ying for their interest in my work. I also appreciate the help of Beverly Pyle in typing the manuscript and of Andrew Tung and T. C. Ma in producing the graphs. REFERENCES BENJAMIN, T. B., and URSELL,F. (1954). The stability of the plane free surface of a liquid in vertical periodic motion. Proc. Roy. SOC.London, Sec. A 225, 505-515. BENJAMIN, T. B. (1967). Instability ofperiodic wave trains in nonlinear dispersive systems. Proc. Roy. SOC. London, Sec. A 299, 59-75. CRAPPER, G. D. (1957). An exact solution for progressive capillary waves of arbitrary amplitude. J. Fluid Mech. 2, 532-540. DAVIS,R. E., and ACRIVOS, A. (1967). The stability of oscillatory internal waves. J. Fluid Mech. 30, 723-736. HASSELMANN, K. (1967). A criterion for second-order nonlinear wave stability. J. Fluid Mech. 30, 737-741. HUNT,J. N. (1961). Houille Blunche 4, 515. PHILLIPS, 0. M. (1966). “The Dynamics of the Upper Ocean.” Cambridge Univ. Press, London and New York. A. R. (1920). Stationary waves in water. Philos. Mag. (6), 40,97-110. RICHARDSON, WHITHAM, G. B. (1966). Nonlinear dispersion of water waves. J. Fluid Mech. 27, 399. YIH, C.4. (1957). On stratified flows in a gravitational field. Tellus 9, 220-227. YtH, C.4. (1965). “Dynamics of Nonhomogeneous Fluids.” MacMillan, New York. YIH, C.4. (1974). Progressive waves of permanent form in continuously stratified fluids. Phys. Fluids 17, 1489-1495.
NOTE ADDED IN PROOF We note also (seep. 391) that for a primary wave train with basic wave number k, the higher harmonics (in the velocity potential, the velocity, the surface form, etc.) are unstable even without considering the increase of the wave velocity with amplitude. When the primary waves are made stationary the Froude number based on the wave velocity of the linear theory is such that m = 0 and m’ = k, as mentioned after (2.55),and hence the wave train is stable for the basic harmonic, with wave number k, of the waves. But for the higher harmonics, with wave numbers 2k, 3k, etc., a nonzero m exists and those harmonics therefore cause instability. Indeed, the growth rate for the second harmonic (with wave number 2k) is proportional to the square of the amplitude of the basic harmonic.
This Page Intentionally Left Blank
Author Index Numbers in italics refer to the pages on which the complete references are listed.
Bridgman, P.W., 346,366 Brock, R. R., 102, I12 Bullard, E. C., 120, 136, 177 Bulson, P. S., 14, 112 Burns, J. C., 87, 89, I22 Busse, F. H., 137, 173, 177
A Abramson, H. N., 351, 365 Acheson, D. J., 170, I78 Acrivos, A., 370, 372, 393, 418, 419 Adidam, S. R., 187, 191, 192, 193, 226, 227, 252,307 Alter, B. E. K., 352, 365 Andrade, E. N. DaCosta, 314,365 Arthur, R. S., 50, 51, 111
C
Campbell, J. D., 353, 366 Casey, T. B., 63, I12 Chandrasekhar, S., 130, 174, 177 Babcock, H. D., 124,176 Charrett, D. E., 190, 193, 195, 296, 298, 300, Babcock, H. W., 124, I76 304, 307 Backus, G . E., 136,176 Chen, D. T., 52, 53, 114 Ballou, J. W., 333, 365 Chern, J. M., 206, 304 Banner, M. L., 11, 12, 70, 99, 110, 111, 116 Childress, S., 173, I77 Barber, N. F., 39, 53, 61, 111, I15 Christie, D. G., 360, 366 Barnett, R, L., 206, 304 Batchelor, G. K., 92, 132, 142, 148, 111, 176, Chu, V. H., 34, 112 Clark, D. S., 351, 366 I77 Clyde, D. H., 226, 304 Beck, R. F., 107, 110, 1 I I Cohn, M. Z., 296, 307 Behannon, K. W., 122, 179 Colburn, D. S., 122, 180 Bell, J. F., 309, 352, 365 Coleman, P. J., 122, 180 Benbow, J. J., 323, 327, 334,365 Collins, J. I., 115 Benjamin, T. B., 31, 90,92,95, 102, 111, 370, Coulomb, C. A., 316, 366 371, 382, 384, 386, 387, 389, 390, 419 Cowling, T. G., 135, 177 Benney, D. J., 77, 95, 112 Craik, A. D. D., 16, 83, I 1 2 Betts, P. L., 112 Crapper, G. D., 20, 33, 45, 48, 59, 69, 122, Biermann, L.,148, 180 385,419 Birkemeier, W.A., 11 7 Cross, J. J., 106, I16 Blythe, P. A., 112 Curtis, C. W., 352, 365 Bodner, S. R., 363, 366
B
Boltzmann, L., 316, 366 Braginskii, S.I., 136, 144, 147, 154, 159, 162, 166, 171, 172, 173, 177 Brard, R.,107, 112 Bretherton, F. P., 18, 29, 35, I f 2 42 1
D Dafalias, Y.F., 206, 304 Dagan, G., 13, 70, 109, 112
422
Author Index
Dalrymple, R. A., 88, 92, 94, 112, I 1 7 Davies, R. M., 329, 366 Davis, E. D. H., 353, 366 Davis, L., 122, 180 Davis, R. E., 370, 372, 393, 418, 419 Dean, R. G., 94, I I2 Delachenal, M. B., 98, I I 2 De Witt, T. W., 322, 367 Dienzer, W., 167, 177 Dillon, R. F., 146, I77 Dolginov, S., 122, 177 Donnell, L. H., 349, 366 Douch, L. S., 353,367 Draper, L., 25, I 1 2 Drazin, P. G., 78, I 1 2 Dressler, R. F., 101, 113 Drobyshevski, E . M., 133, 177 Drucker, D. C., 192, 220, 304, 305 Dubreil-Jacotin, M. L., 92, 113 Dudgeon, C. R., 26,1 I3 Dupuis, G., 206, 304 DuvaH, G. E., 346,366 Duwez, P.E., 350, 351, 366, 368 Dyal, P., 122, I80
E Eltayeb, I. A., 171, 172, 177 Emery, K. O., 51, 116 Evans, D. V., 74, 75, 113 Evans, J. T., 14, 15, 45, 113 F Farell, C., 109, 113 Fenton, J. D., 20, 79, 99, 113. 115 Ferry, J. D, 336, 366 Fitzgerald, E. R., 336, 366 Foulkes, J., 196, 305 Francis, J. R. D., 26, 113 Fredsde, J. 79, 80, 82, I 1 3 Freeman, N. C., 95, 113 Furth, H. P., 172, 177
Gangadharaiah, C., 218, 227, 300, 308 Gargett, A. E., 12, 18, 52, 113 Garrett, C. J. R., 29, 35, 112 Gellman, H., 129, 136, I77 Gibson, R. D., 137, 160, 177, 181 Goldsmith, W., 351, 366 Goldstein, M . E., 71, 113 Golitsyn, G. S., 142, I77 Gouyon, R., 92, 113 Graham, B. B., 72, 113 Graham, E. W . , 72, I I3 Grant, M. A., 98, 113 Green, J. L., IS, 113 Green, T., 104, 105, 106, I I3 Groves, G . W, 61, 116 Gubbins, D., 125, 178 Guven, O., 109, I I3
H Hales, L. Z., 50, 113 Harper, R. C., 322, 367 Hasselmann, K. F., 35, 61, 68, 113, 116, 370, 391, 418, 419 Haug, E. J., 206, 305 Hayes, W. D., 29, 32, I I3 Hemp, W. S., 201, 305 Herbich, J. B., 50, 113 Herzenberg, A., 137, I78 Heyman, J., 191, 225, 305 Hide, R., 120, 123, 170, 178 Higgins, G. H., 171, 172, 178 Hill, R., 218, 227, 287, 300, 307, 308 Hillier, K. W., 331, 333, 334, 339, 346, 366 Hodge, P. G., 292, 305 Hogben, N., 11, I I3 Holliday, D., 46, I 1 3 Hopkinson, B., 355, 366 Hopkinson, J., 355, 366 Howard, L. N., 78, 1I 2 Howe, M. S., 104, 114 Huang, N.E., 52, 53, 114, 117 Hughes, B. A., 12, 52, 61, 113, 114 Hunt, J. N., 418, 419 Hunter, S . C., 336, 353, 366
G Gadd, G. E., 110, 111, 113 Gailitis, A., I77
I Ibbetson, A., 150, 178 Irwin, L. K., 348, 367
Author Index J Jahsman, W, 353, 366 James, H., 329, 366 James, I. D., 13, 114 Jeffreys, H. J., 101, 114 Jepps, S. A., 167, 178 Johns, D. J., 26, 114 Johnson, J. W., 50, 114 Johnson, R. S., 95, I 1 3 Jolly, J. P., 102, 1 I4 Jones, D. E., 122, 180 Jones, D. S., 72, 114 Jonsson, I. G., 24, 40,I 1 4
K Kaliszky, S., 226, 305 Kang, S. W., 106, 113 Kazakia, Y., 112 Kennedy, G. C., 171, 172, 178 Kenyon, K., 6 1, 1 14 Kiepenheuer, K. 0.. 167, 178 Killeen, J., 172, 177 King, R., 26, I 1 4 Kinra, V. K., 363, 366 Kirmser, P. G., 206, 305 KO, D. R. S.,52, I14 Kolsky, H., 331, 336, 337, 343, 344, 345, 346, 351, 353, 355, 357, 358, 359, 360, 363, 366, 367,368 Kovasznay, L. S. G., 178 Kraichnan, R. H., 130, 148,178 Krause, F., 144, 146, 163, 164, 165, 166, 178, 181
L Lackenby, H., 106, 114 La Ford, E. C., 51, 116 Lake, B. M., 52, 114 Lamb, H., 93,98, 114 Lamblin, D. O., 194,305 Lanczos, C., 38, 114 Landahl, M. T., 100, 114 Le Blond, P. H., 51, 114 Lee, E. H., 346, 351, 367 Lee, J. P., 346, 351, 362, 367
423
Lehnert, B., 168, 178 Leighton, R. B., 167, 178 Lepping, R. P., 122, I79 Lerche, I., 154, 178 Lethersich, W., 322, 335, 367 Lewis, J. E., 52, 114 Liepmann, H. W., 142, I78 Lifshitz, J., 342, 367 Lighthill, M. J., 31, 32, 33, 71, 79, 90, 142, 111, 114, 178 Lindholm, U. S., 353,367 Longuet-Higgins, M. S., 10, 13, 16, 20, 33, 35, 36, 61, 62, 65, 67, 68, 69, 99, 114, 115
L o r t i D., 136, I78 Lowe, P. G., 218, 226, 227, 255, 256, 297, 298, 305 Lowes, F. J., I78 Ludwig, D., 38, 115 Luke, J. C., 28, 115 Lumley, J. L., 148, 178
M McKee, W. D., 56, 57, 58, 59, I 1 5 Malkus, W. V. R., 170, 174, I79 Mallory, J. K., 62, I I5 Malvern, L. E., 351, 367 Marcal, P. V., 190, 305 Markovitz, H., 322, 367 Martin, J. B., 190, 306 Masur, E. F., 206, 208, 298, 305 Maunder, E. W., 167, I79 Mayeda, R., 194,305 Medwin, H., 104, 105, I I3 Mei C . C., 34, I 12, I15 Melchers, R. E., 218, 220, 226, 227, 255, 256, 297, 298, 305, 307 Miche, M., 98, 115 Michell, A. G. M., 184, 305 Miklowitz, J., 360, 367 Miles, J. W . , 115 Miller, G. R., 61, 1 15, I I6 Moffatt, H . K., 122, 125, 142, 148, 149, 151, 161, 169, 170, 179 Moffett, M. B., 345, 367 Moiseev, N. N., 92, 1 15 Moore, W. L., 1 I5 Morgan, C. W., 72, 115
Author Index
424
Morgan, J. D., 114 Morley, C . T., 208, 220, 297, 298, 405 Mroz, Z., 197, 198, 200,206,208, 220, 226, 298, 306,308 Munk, W. H, 61, 115,116
N Nagarajan, S., 148, 178 Nagtegaal, J. C., 201, 306 Nece, R. E., 14, 115 Ness, N, 122, 178 Noda, E . K., SO, 115 Nolle, A. W., 333, 367 0 Oi, K., 360, 367 Okajima, A., 26, 116
P Paquin, J. E., 104, 105, 113 Parfitt, G. G., 334, 367 Parker, E. N., 124, 125, 131, 144, 147, 149, 153, 154, 163, 164, 165, 167, 168, 179 Parker, R. L., 131, 279 Parkes, E. W., 306 Pearcey, T., 38, I I5 Pearson, J., 359, 368 Peregrine, D. H., 13, 14, 50, 56, 57, 58, 74, 79, 88, 89, 98, 102, 103, 110, 1 1 5 , I16 Peters, A. S., 102, 103, 116 Phillips, J. W., 360, 367 Phillips, 0. M., 11, 12, 31, 34, 42, 52, 53, 70, 99, 104, 105, 110, 111, 116, 370, 388, 419 Pichakhchi, L. D., 136, 180 Piddington, J. H., 163, 167, 180 Plass, H. J., 351, 365 Pohle, F. V., 101, I13 Powers, W. H., 116 Prager, W., 185, 186, 187, 188, 190, 192, 193, 194,196,197,199, 201,205,206,207,208, 298, 304, 304, 305, 306, 308 Proctor, M. R. E., 134, 174, 175, 179,180 Prosser, M. J., 26, 114 Puri, K. K., I ! 7
Q Quimby, S. L., 334, 343, 367
R Rader, D., 357, 358, 367 Radler, K.-H., 151, 152, 164, 180, 181 Rakhmatulin, H . A., 350, 367 Ramberg, W., 348, 367 Randall, R. H., 334, 367 Rao, V. S., 14, 115 Reitman, M. I., 226, 306 Ribner, H. S., 116 Richardson, A. R., 372, 419 Richey, E. P., 14, 115 Rinehart, J. S., 359, 368 Ripperger, E. A., 348, 351, 365, 368 Rivlin, R. S . 364, 368 Roberts, D. K., 361, 368 Roberts, G. O., 120, 144, 165, 180 Roberts, P. H., 125, 130, 133, 146, 152, 166, 167,170,171,172, 173, 175, 176,177.278 180 Rose, F. C., 334, 367 Rosenbluth, M . N., 172, 177 Romany, G. I. N., 186, 187, 190, 191, 192, 193, 195, 196, 197, 198, 199, 200, 204, 205, 208, 218, 220, 226, 227, 231, 234, 252,287,291,296,298, 300, 302, 304, 306, 307,308 Rupert, V. C., 1 15, I 17
S Sacchi, G., 226, 308 SaNman, P. G., 148, 180 Sarpkaya, T., 100, 101, 104, 116 Save, M. A., 194, 208, 226, 298, 305, 308 Savitsky, D., 105, 116 Schatten, K. H., 122, 179 Schluter, A., 148, 180 Schwartz, L. W., 20, 33, 45, 116 Shamiev, F. G., 220, 306 Shearer, J. R., 106, 116 Shearman, A. C., 357, 358, 359,367 Shemdin, 0. H., 70, 76, 79, 116 Shepard, F. P., 51, 116 Sheu, C. Y., 196, 206, 308 Shield, R. T., 188, 190, 192, 194, 206, 209, 220, 298, 304, 305, 306, 308
Author Index Silcock, G., 100, I16 Silverman, S., 333, 365 Skougaard, C., 24, 40, 114 Sleath, J. F. A., 16, 116 Smith, E., 122, 180 Smith, J. R., 52, 53, 114 Smith, R.,56, 57, 58, 74, 79, 88, 89, 116, I I7 Snodgrass, F. E., 61, 115, 116 Sonett, C. P., 122, 180 Sono, C. J., 115 Soward, A. M., 125, 146, 154, 160, 161, 169, 173, 174, 176, 177, 181 Sridhar Rao, J. K., 297, 308 Steenbeck, M., 163, 164, 165, 166,178, 181 Sternglass, E. J, 352, 368 Stewart, R. W, 10, 35, 36, 51, 61, 63, 65, 67, 68, 114, 115 Stewartson, K., 171, 172, 173, 176, 180 Stix, M., 167, 177, 180 Stokes, G. G, 20, 116 Stuart, D. A., 352, 368
T Tang, C. L., 51, 114 Tanida, Y., 26, I I6 Tatinclaux, J.-C., 107, 110, I I6 Taylor,G. I., 15, 36, 79,91, 116, 148, 181, 350, 368 Taylor, J. B., 171, 181 Taylor, J. E., 206, 306 Thakkar, M.C., 297, 308 Thompson, P. D., 87, 88, 116 Thomson, J. A., 1 I7 Tough, J. G., 160,181 Tritton, D. J., 150, 178 Tsai, Y . M., 360, 368 Tsao, S., 93, 94, I16 Tung, C.-C., 52, 53, 114, 117 Tupper, S. T., 346, 367
425
Varley, E., I 1 2 Velthuizen, H. G. M.,83, I I7 Venezian, G., 137,181 von Karman, T., 350, 368 von Kusserow, H. U., 167, I77 Vincent, C. E., 26, 52, 1 1 7 W
Walther, H., 334, 368 Wang, J. D., 24, 40, 114 Warwick, J., 122, 181 Watanabe, Y., 26,116 Watson, H. L., 293, 308 Wegel, R. L., 334, 368 Wehausen, J. V., 106, 109, 117 Weiss, N. O., 125, 131, 132, 181 Wells, A. A., 360, 368 West, B. J., 1 1 7 Whang, Y. C, 122, 179 Whitham, G. B., 18, 19, 27, 28, 29, 32, 33, 34,117, 370, 371,382, 386,389,419 Wiegel, R. L., 14, 117 Wilkinson, I., I78 Williams, J. A., 14, 117 Woltjer, L., 181 Wood, R . H., 220, 308 Wu, J., 76, 11 7
Y Yavorsky, P., 322, 367 Yeakley, L. M., 348, 368 Yeroshenko, Y., 122, 177 Yevjevich, V., 102, 114 Yih, C.-S., 78, 84, 89, 90, 100, 117, 372, 418, 419 Yoffe, E. H., 359, 368 Yuferev, V. S., 133, 177
U Ursell, F., 371, 419
2 V
Vainshtein, S. I., 125, 181 van Wijngaarden, L., 83, 117
Zapus, L. J., 322, 367 Zel’dovich, Y. B., 125, 136, 181 Zener, C., 334, 364,367, 368 Zhuzgov, L., 122, 177
This Page Intentionally Left Blank
Subject Index A
C
Aeronautical Research Council, 7 Agulhas current, 62 Alfven’s theorem, magnetic field and, 128 Antidynamo theorems, 135-137 Atmosphere, friction layer of, 6 Axially symmetric supports and loading, optimal flexure fields and velocity for, 220-226 Axisymmetric mean fields, dynamo equations for, 152-154 Axisymmetric turbulence, pseudotensors in, 150-152
B Beam layouts, optimal and nonoptimal, 186 Beam systems, in floors and roofs, 186 Benard convection, between horizontal planes, 133 Blast wave, measurements of, 5 B-lines, diffusion of in magnetic fields, 130-135 Boltzmann’s principle, in viscoelastic behavior, 316-322 Booleian algebra, 2 Braginskii’s theory advantages of, 162 of nearly axisymmetric fields, 154162 Breakwater, bubble or pneumatic, 1 4 1 5 Breakwater current, 15 Brittle fracture, stress waves in, 360-362 Brittle solids crack propagation in, 359-360 fracture of, 3 5 4 3 5 5 427
Capillary-gravity waves, properties of, 69 Capillary waves, generation of near gravity wave, 69 Cavendish Laboratory, Cambridge, 1, 4, 6-7 Clamped boundaries outer ends and branches of, 229-230 properties of solutions for, 227-231 Coastal waves, 12-13 see also Waves Compliance or deflection constraints, in optimal load transmission by flexure, 204205 Convection, in magnetic fields, 130-135 Coriolis forces, 139 dynamo and, 168-170 helicity and, 164 waves influenced by, 168-170 Cost distribution design value of cost in, 193 partially prescribed, 195- 196 unspecified, 190-195 Cost functions, specific, 189- 191 Cost gradient surface, 192 Cowling’s theorem, 135-137 CQR anchor, 5 Crack propagation, in brittle solids, 359-360, 363 Curie point, magnetization and, 120 Current(s) see also Waves Agulhas, 62 Antarctic circumpolar, 62 generation of, 16 jetlike, 58, 62 large-scale, 17-70 rip, 50-51 shear flow and, 53 slowly varying, 26-39
Subject Index
428
small-scale, 70-75 steady, varying across stream, 53-63 steady, varying with stream distance, 39-53 uniform, 18-26 varying with depth, 76-103 varying with stream distance, 39-53 vertical accelerations in, 63-70 water waves and, 9-1 11 wave steepness and, 67
D Depth grillages, optimization of, 187 Dirac delta function, 197 Differential rotation, toroidal field generation by, 134-135 Displacements, generalized, 187 Distortion, in magnetic fields, 130-135 Dynamic fracture, experimental studies in, 354-363 Dynamo u2 version with (x constant, 163-164 Coriolis forces and, 168-170 global behavior of, 166-168 local behavior of, 166 rotor, 137-138 with symmetry, 164-165 Dynamo action, in electric field, 135-137 Dynamo equations analytical and numerical solutions for, 163-168 for axisymmetric mean fields, 152-154 for nearly rectilinear flows, 158-160
E Earth, magnetic field of, 120 Ekman spiral, derivation of, 6 Elastic design compliance constraints in, 201 deflection constraints in. 201 strength constraints in, 201 Elastic structure, cost of, 201-202 Electromotive force, random velocity field of, 139-154 Experiment, role of in solid mechanics, 309-365
F Finite-amplitude waves, 91-99 equations for, 92-93 Froude number for, 95,97 highest, 98-99 solutions for, 93-99 Flexure, optimal load transmission by, see Optimal load transmission by flexure Flow in channels, waves on, 102-103 Flux expulsion, by flows with closed streamlines, 131-133 Force-free fields, 129-1 30 Four-dimensional geometry, 2 Fracture, dynamic, see Dynamic fracture Froude number defined, 371 instability and, 371 ship waves and, 109 stability and, 100-101 for waves with small-scale currents, 90, 95, 97
G Generalized displacements, 187 Generalized gradient operator, 189-190 graphical representation of, 189 Generalized loads, 187 in given location, 196-197 Generalized reactions, of unspecified magnitude and location, 199-201 Generalized stresses, 187 General optimality criteria, elastic design and, 201-206 see also Optimal flexure fields Gravity waves generation of capillary waves near, 69 instability of, 386-391 Grillagelike continua, optimal. 184-185 Group velocity, of waves, 9 I
H Helicity, Coriolis forces and, 164 Hinges, optimal location of, 205-206
429
Subject Index Hooke’s law deviations from, 311-313 three-dimensional form of, 310, 317 Hydraulic breakwaters, 14-15 Hydrodynamic stability theory, 78
of induction equation for nearly axisymmetric magnetic fields, 154-155 Large-scale currents, 17-70 see also Current(s) Loads, generalized, 187 Lorentz forces. 139
I M Iceberg research, 5-6 Infinitesimal waves, 77-91 see also Waves approximate solutions for, 86-89 critical layer in flow of, 82-86 group velocity of, 91 upstream propagation of, 90 velocity profiles for, 79-82 wave motion equations for, 77-79 Instability see also Surface waves, instability of formal demonstration of, 377-382 of gravity waves, 386-391 Instability wave, 72 Interfacial waves, instability of, 394408 Internal waves instability of, 393-419 resonance across modes of, 417-418 stationary interfacial, 394408 Inviscid flow, stability of, 100
J Joints or connections, of nonzero cost, 197-198
K Kelvin-Voight elements, in solid mechanics, 320-321 Kinematically admissible displacements and strains, 188 Kinematic dynamo problem, idealization of, 125-127
L Lagrangian transformation arranged, 28-34
MAC (Magnetic, Archimedian, and Coriolis) waves, 171, 173 see also Magnetostrophic waves Magnetic energy, mean flow effect and, 175 Magnetic field Alfven’s theorem and, 128 asymmetric, 127 axisymmetric, 150-154 convection, distortion, and diffusion of B-lines in, 130-135 Cowling’s theorem and, 135-137 differential rotation in, 122 dynamic effects in, 168-176 of Earth, 12e121 effective, 157- 158 force-free, 129 generation of by fluid motion. 119-176 growth and velocity of, 174175 kinematic dynamo problem and, 125-127 Lagrangian transformation of induction equation for, 154-155 magnetok inematic properties and, 125- 130 mean flow equilibrium in, 174-176 natural decay modes and force-free fields in, 129-130 nearly axisymmetric, 154-162 nearly rectilinear flows and, 157-158 poloidal and toroidal decomposition of, 127 representations of, 127-128 rotation of, 121-122 self-equilibration and, 168-176 of Sun, 123-124 two-dimensional, 128 Woltjer’s invariant and, 128 Magnetic flux expulsion of, 131-133 topological pumping of, 133-134 Magnetic flux rope, stretching and diffusion in, 131
Subject Index
430
Magnetization, temperature and, 120 Magnetokinematics, 125-130 Magnetostrophic balance, 170 Magnetostrophic flow, 170-171 Magnetostrophic waves, excitation of by unstable stratification, 171-174 Mass flow, in wave theory, 68 Mass transport, in uniform wave train, 16 Maxwell elements, in solid mechanics, 320 Mean electromotive force random velocity field generation of, 139- 154 strong diffusion limit and, 141-145 Mean flow effects dynamo equations and, 152-154 magnetic energy and, 175 Monash University, 186 N
Nearly axisymmetric magnetic fields, 154-162 Nearly axisymmetric systems, 155-157 Sowards approach in, 160-161 vs. two-scale systems, 161-162 Nearly rectilinear flows, dynamo equations for, 158-160 Nonaxisymmetric optimal flexure fields, literature on, 226-227 Nonzero cost, generalized loads and connections of, 197-198 0
Oil tankers, breaking of, 62 Optimal elastic designs, coincidence with plastic design, 203 Optimal flexure fields ah,b6, and y6 branches in, 290 axially symmetric supports and loading in, 220-226 basic geometrical properties in, 206-226 branches and junctions in, 219-220 branches containing a and J/’ fields only, 246-250 circular arc domains in, 267-269 clamped boundaries in, 226-246 combined bending and shear in, 293-294
combined bending and torsion in, 292-293 combined upward and downward loads in, 296 of constrained geometry, 299-303 convex polygonal domains and, 239 defined, 208 6 field construction in, 285 66 branches in, 286-289 domains of bounded sets in, 239-241 domains of half-space and bounded set in, 243-244 domains with curved boundaries in, 244246 examples of, 239-246 examples of a and fl fields in, 252-272 examples of 6 fields in, 280-282 free-edge problems in, 299 further developments of, 292-299 y and 6 fields in, 272-2d1 grillage with main and secondary beams in, 300-303 inconsistent results in, 296-298 junctions in, 291 load transmission to simply supported boundaries in, 272-273 mixed boundary conditions in, 246-292 optimal grillages for alternate loads in, 296 optimal moment-curvature rate relations in, 206-209 and other types of interfield boundaries, 291-292 properties of S- and T-type regions in, 209-210 quadrilateral domains in, 255-267 for rectangular domains, 253-254 R-type regions in, 210-214 solutions for internal supports in, 294295 for square domains, 252 straight-line and point supports for, 242-243 straight-line-segment domains in, 271-272 topographical properties of, 218-220 triangular domains in, 255-256 T-shaped domains in, 270 Optimal load transmission by flexure, 183-303 compliance or deflection constraints in, 204-205 discontinuous cost functions in, 198-199
Subject Index general optimality criteria and, 201-206 optimal flexure fields in, 206-226 static-kinematic optimality criteria in, 187-201 strength design in, 202-204 superposition principle in, 201 Optimal moment-curvature rate relations, derivation of, 206-209 Optimal segmentation, partially prescribed cost distribution and, 199 Optimal velocity fields, for axially symmetric supports and toading, 221-226 Organic glasses, behavior of, 336-337 Orr-Sommer field equation, 78
P Partially prescribed cost distribution fixed segmentation and, 195-196 optimal segmentation and, 199 Perturbation equations, for waves, 36-39 Planets, magnetic field of, 122 Plastic design optimal elastic design and, 203 static-kinematic optimality criteria for, 187-201 Plastic-wave front, propagation of in wires, 351-352 Plastic waves, propagation of, 349-354 Polyethylene, viscous behavior of, 336 Polyethylene blocks, spherically diverging pulses in, 343 Polyethylene rods, pulse propagation in, 338-340 Polymeric materials, as simple model solids, 336 Polymethylmethacrylate, viscous behavior of, 336 Polystyrene, stress pulses in, 345 Polyvinyl chloride, stress pulses in, 345 Prager-Shield criteria, 191-192, 202 Primary flow, of surface waves, 372-375 Progressive waves, vs. stationary, 418 Pseudotensors in axisymmetric turbulence, 150-152 decomposition of, 144 diffusion limit and, 143 evaluation of for random wave field, 145- 147 Pulse propagation, in rods, 338
43 1 R
Rqdiation stress, water waves and, 10 Random velocity field, mean electromotive force generated by, 139-154 Random wave field, evaluation of pseudotensors for, 145-147 Rayleigh equation, 78 Rays, focus of, 38 Ray theory, wave patterns and, 37-38, 44 Reinforced concrete slabs, 186 Rip currents, generation of, 5&51 see also Current(s) Rotation differential, see Differential rotation magnetic field and, 122 Rotor dynamos, 137-139 Royal Society, 8 R-type regions, in optimal flexure fields, 21&214 Rubber plastic vs. shock waves in, 345-346 tensile shock waves in, 345-349
S Salt solutions, stability of, 4 Sea waves, 11-12 see also Waves group velocity of, 50 Segment boundaries, optimal location of, 205-206 Ship resistance, physical analysis of, 107 Ship’s motion, resistance to, 107-108 Ship waves, 15, 10&111 Shock waves, structure of, 6 Slowly varying currents, waves on, 26-39 Small-scale currents, 70-75 finite-amplitude waves and, 91-99 stability and, 99-102 Solid, constitutive relation of, 310 Solid mechanics brittle fracture in, 360-362 dynamic fracture in, 354-363 Maxwell and Kelvin elements in, 320-322 plastic wave propagation in, 349-354 relaxation time in, 320 role of experiment in, 309-365 spalling in, 354-358
432
Subject Index
system inertia in, 315-316 tensile shock waves in, 345-349 types of experimental research in, 310-311 viscoelastic behavior in, 312, 316-322 viscoelastic response determination in, 322-345 Spalling, 354-358 Specific cost function, 189 homogeneous, 191 Specific internal work, 188 Specific power, 188 Split Hopkinson bar, 353 Stability Froude number and, 100-101 of inviscid flows, 100 Stability problem, for surface waves, 375-377 Standard linear solid, 320 Statically admissible generalized loads and strains, 188 Static continuity conditions, 188 Static-kinematic optimality criteria, 187-201 Stationary waves progressive waves as, 418 in continuously stratified fluid, 408-417 instability of, 394-417 Steepness, in waves, 67-70 Stokes wave expansion, 21 Stokes wave solution, third-order, 32 Strains, generalized, 187 Stratification, magnetostrophic waves and, 171- 174 Strength design, Prager-Shield condition in, 202 Stresses, generalized, 187 Stress relaxation, 312 Stress space, 189 Stress-strain relationship, time dependence in, 313-314 Structural optimization, trusslike continua in, 184 Structural theories, parameters in, 187-188 S-type regions, in optimal flexure fields, 209-210 Sun magnetic field of, 123-124 motion in convection zone oc 139 Sunspot activity, magnetic field and, 123 Sunspot formation, “cyclonic events” and, 125
Supports, optimal location for, 205-206 Surface shear waves, 97-98 Surface tension, in surface wave instability, 385-386 Surface wave amplitude, error in, 25 Surface wave instability, surface tension in, 385-386 Surface waves, 369-393 see also Water waves; Waves primary flow and, 372-375 resonance and, 37 1 stability problem formulation for, 375-377 stability vs. resonance in, 379 surface tension and, 385-386 wavelengths and growth rates of unstable modes in, 382-384
T Tankers, breaking of by waves, 62 Taylor constant. 170-171 Tensile shock waves, in rubber, 345-349 Toroidal magnetic field, generation of by differential rotation, 134-135 Trusslike continua, optimal, 184-185 T-type regions, optimal flexure fields and, 209-210 Turbulence axisymmetric, 150-152 wave action and, 103-106 weak diffusion limit and, 147-150 wire gauge and, 1 Turbulent transfer, in atmosphere, 6
U Uniform currents, waves on, f 8-26 Uniform energy dissipation principle, 192 Unspecified cost distribution alternate loads in, 193-194 multicomponent systems and, 1 9 4 195 single load system and, 190-193
V
Velocity profiles, for infinitesimal waves, 79-82
Subject Index Vertical acceleration, in currents, 63-70 Viscoelastic behavior, 312, 316-322 Viscoelastic properties apparatus for measurement of, 323 torsional oscillation apparatus for, 326 Viscoelastic response, experimental determination of, 322-345 Viscoelastic solids plastic vs. shock waves in, 345-346 response of to volume changes, 342 stress-strain relations in, 318 stress wave propagation in, 329-332 wave propagation in, 333-334 Volume viscosity effects, in viscoelastic solids. 432 Vortex sheet, waves in, 73-75 Vorticity-transfer theory, 6
W
Water waves see also Waves currents and, 9-1 1 1 notation for, 16-17 Wave-action conservation equation, 35 Wave-action density, 29 Wave-action Rux, 29 Wave motion, local solution for, 43 Wave number, 18-19 Wave train, stability of, 31 Waves ansatz for, 37 averaged equations of motion for, 34-35 averaged Lagrangian and, 28-34 breaking of near ships, 107 capillary-gravity, 32, 35, 46 caustic and, 37 coastal, 12-13 Coriolis forces and, 168-170 deep-water gravity. 32-33 eikonal equation for, 37 finite-amplitude, 45, 91-99 A 6 on Row in channels, 102-103 8 7 C 8 focus of rays and, 38 D 9 E
O
F
1
G 2 H 3 1 4 J 5
433
friction laws and, 104 generation of current from, 16 gravity, 32. 34 group velocity of, 91 highest, 98-99 infinitesimal, 34, 77-91 instability type, 72-73 internal, see Internal waves jetlike flow and, 58 large-scale currents and, 17-70 long-interval, 52 long-traveling, 51 mass flow from, 68 nonlinear dispersive, 27 nonlinear properties of, 11 perturbation equations and asymptotic expansions for, 36-39 pure capillary, 33 radiation stress and, 35 ray theory for, 37-38, 44 rip currents and, 50-51 in rivers and channels, 13-14 sea, see Sea waves ship, 15, 106-111 on slowly varying currents, 26-39 stationary, see Stationary waves steepness of, 67 Stokes third-order solution for, 32 stopping point for, 48 stopping velocity for, 46, SO steady current and, 39-63 surface, see Surface waves on “top-hat” jet, 74 transport equations for, 37 turbulence and, 103-106 in uniform currents, 18-26 on vortex sheet, 73-75 Wavy Rows, instability of, 384-385 Weak diffusion limit, turbulence and, 147-150 Wimshurst machine, 5 Wind stress, surface drift from, 11 Wind waves, energy spectrum for, 52-53 Wing, pressure distribution on, 6 Wire gauge, turbulence and, 1 WoltJer’s invariant, magnetic field and, 128
This Page Intentionally Left Blank
