Editors' Note

In 1985, a symposium in honor of Professor Chia-Shun Yih was held in Ann Arbor, Michigan. The present volumes are part of the outcome of that symposium, of which the organizing committee appointed us editors of these volumes. Several other former students of the author have also helped in our endeavor. In particular, the efforts of Professors Walter R. Debler and Timothy W. Kao are gratefully acknowledged.

Since Professor C. C. Lin introduced the author to the use of mathematics in the study of fluid mechanics in the summer of 1947 at Brown University, we have asked him to write the preface for these volumes. The preface is followed by three articles by the author's friends, on him and his work. Then, in each volume, after the corrigenda, prepared by the author himself for the papers contained therein, the selected papers are presented in the order indicated bathe table of contents. At the end of Volume II we have included a curriculum vitae and a list of publications of the author, followed by the corrigenda for papers not included in these volumes.

Apart from the research papers, Yih also published the book Dynamics of Nonhomogeneous Fluids, Macmillan, New York, 1965, which was revised in 1980 and published under the title Stratified Flows by Academic Press. This book has been the standard reference book for researchers studying the behavior of stratified fluids. Yih's Fluid Mechanicswas published by McGraw-Hill Company of New York in 1969, and was revised in 1979 and republished by West River Press of Ann Arbor. This book is often referred to, and is still used as a textbook in some American and European universities.

Looking over the titles of Yih's papers, one can hardly escape the impression that a great many of them deal with the dynamics of fluids stratified in density, entropy, viscosity, thermal conductivity, electrical conductivity, circulation, or else magnetic circulation. This bears testimony to his fascination with the effects of heterogeneity of all sorts in fluid flows, which provide a unifying thread through a great many, perhaps the majority, of his papers.

As a teacher, he is known generally and especially among his thirty-five research students, as a source of inspiration. His enthusiasm for fluid mechanics provides a contagious stimulation to his students. His presentation of some of the beautiful theories of fluid mechanics often took on a magician's touch, as when he gave the transformation with which a large-amplitude wave motion of a stratified fluid can under certain conditions be shown to be governed by a linear partial differential equation -- or even when he presented the classical theorems of circulation and vorticity. His enthusiasm and his love of beauty in Nature and in science will be remembered by those of us who know him.

Sung-Piau Lin
W. Michael Lai




Preface

The editors of the present volumes have asked me to provide some comments on the significance of the contributions of Professor Chia-Shun Yih to the field of fluid mechanics. I am most happy to do so. Yih's contributions to the literature of fluid mechanics have been important and extensive. Besides some general comments, I shall also make some specific comments which are limited to a few salient points related to a small portion of his work.

In this volume, Yih's papers are collected into five parts:

  • A. Stratified Flows and Internal Waves,
  • B. Theory of Hydrodynamic Stability,
  • C. Gravity Waves
  • D. Jets, Plumes and Diffusion,
  • E. General

    • Of these, the first -- stratified flows and internal waves -- is clearly afield that Yih has made his own. But he is at heart a naturalist and an engineer, and his contribution to the literature of fluid mechanics have been quite broad, and span over a variety of physical phenomena and a wide area of applications. The diversity of his interest may be gleaned from Part E of his papers.

    With his bountiful love for nature, it is not surprising that some of his most significant contributions deal with fluid flows that are ubiquitous in our environment. The field of stratified flows itself is rich in physical phenomena.This richness owes its origin to the interplay of the heterogeneity of the fluid medium and the gravitational field. This fact was noted by Yih in the preface to his book Stratified Flows, 1980, which first appeared in 1965 under the title Dynamics of Nonhomogeneous Fluids. This book, which is primarily based on Yih's original contributions to the field, has since become a classic reference for all serious students and researchers. The reader cannot but feel the author's enthusiasm for the subject, that he has indeed achieved an insight to the intricacies of the subject and that he is eager to share the fruits of his labor with the reader.

    While much of Yih's success owes to his physical insight, it is perhaps equally true that he is always able to use precisely the right mathematical tool, especially for the purpose of enlarging our general perspectives. For example, in the paper Gravity waves in a stratified fluid (A5), he made use of Sturm's second comparison theorem to obtain the result that the phase speed decreases as the wave number increases, in small-amplitude wave motion of a heterogeneous liquid with a free surface. He has also been very successful in introducing ingenious mathematical transformation of variables to enable certain general classes of difficult problems of wavy motions with finite amplitudes to be solved with relative simplicity. Such skillful use of mathematics has great impact on a number of scientific applications, including geophysics and water-quality engineering.

    In view of the importance of these mathematical approaches, it is perhaps appropriate to describe some of them in some detail. In a series of papers starting with the 1958 paper (A3) on the flow of a stratified fluid, he demonstrated the use of a transformation which enables one to cast the equation of two-dimensional steady flow of a stratified fluid into a form which can be rendered exactly linear from suitable choices of upstream conditions. He used a transformed stream function tex2html_wrap_inline26 defined as

    where (u, w) are the components of velocity in the (x, z)directions and $\rho$ is the density. It can then be shown that the governing equation for the stream function is

    $\nabla^2$ + (g z/$\rho_0) = (1/$\rho_0)

    in which g is the acceleration of gravity, $\rho_0 is a reference density and H is the total head, which is a constant along a streamline and therefore a function of only. The functions and are to be determined from upstream conditions and if these functions are linear in tex2html_wrap_inline26 the solutions of the linear governing equation yield exact solutions to large-amplitude motions. As Yih noted in his book, "much cane achieved in this way." Similar transformations have also been given by him for a compressible fluid.

    He was, in fact, able to solve several important classes of problems including those given in Exact solutions for steady two-dimensional flow oaf stratified fluid (A6), and A transformation for non-homentropic flows,with an application to large-amplitude motion in the atmosphere (A7). Many of these solutions have important practical application in geophysics and water-quality engineering. In the paper A class of solutions for steady stratified flows (A12), the use of a more general form of transformation of variables led Yih to an elegant theorem in three-dimensional shallow-water theory. The theorem states that "so long as the shallow water theory is valid, a class of steady stratified flow with a free surface originated from rest can be found corresponding to each irrotational steady free-surface flow of a homogeneous fluid originated from rest." The mapping is by the use of the relationship (u, v, w) = (U, V, W) where (u, v, w) and (U, V, W) are the three components of velocity and is a function of the density $\rho_0.

    The subject of internal hydraulic jumps, of great importance to environmental and hydraulic problems, was first treated by Yih and Guha in 1955 (Al). That paper has been referred to in nearly every paper written on that subject ever since. The phenomenon of hydraulic jump in a rotating fluid was later treated by Yih and coworkers (E11).

    Yih's work is not limited to the field of stratified flows and internal waves. In the field of hydrodynamic stability, one is impressed by the span, in time and scope, of his work. The investigations include results that are applicable generally, such as Stability of two-dimensional parallel flow for three-dimensional disturbances (BI), Stability of unsteady flows or configurations, Parts I and II (B16, B18), andEigenvalue bounds for the Orr-Sommerfeld equation (B17). He has also examined how specific fluid properties may affect the instability of the flows: for example, the instability of an electrically conducting fluid (B3, B4), the instability of a non-Newtonian fluid (B12), the effect of viscosity stratification (B13), and the effect of stratification in thermal conductivity(B23).

    The solutions to many stability problems involved the utilization of new techniques or adapting those in other contexts. Thus, Yih's application of a perturbation technique for the problem of the stability of a liquid flowing down an inclined plane (B9) permitted a straightforward calculation to be made in a complex problem, a result that permitted others to adapt the methodology to other cases in which a long-wave solution would be informative. Also, the exploitation of the Floquet theory for periodic excitations allowed him to explore the instability of time-periodic temperature fields (B 18) or time-periodic flows in a circular pipe (B22).

    While Yih's work on hydrodynamic stability contains impressive components necessary for the academic development of the various subjects treated, much of his work was inspired by the needs in engineering problems. It is indeed the examination of the paper-making process that led him to the papers associated with Rayleigh-Taylor instability: On the instability of stock on a Fourdrinier wire (B8), and Effect of variation of acceleration on free-surface instability (BlO). An inquiry from the Boeing Aircraft Company has been the reason for him to write the paper Wave formation on a liquid layer for de-icing airplane wings (B26).

    As mentioned above, Yih has devoted much of his efforts to the study of stratified flows and internal waves. It is natural that he would also work on surface waves in fluids of homogeneous density. His work on waves in flowing water takes into account the effect of the velocity profile. He gave some comparison theorems for water waves in basins of variable depth in 1976 (C3), and studied waves in channels of various cross sections in 1984 (CS). Fascinated by the effect of geometry on water waves he also gave analytical results for Binnie waves (C4), waves in meandering streams (C6), and edge waves created by along-shore current and a ridge in the sea bed (C7). In recent years, he turned his attention to classical problems in water-wave theory. Four papers (C8, C9,C12 and C13) are concerned with linear or nonlinear water-wave groups. The papers on nonlinear waves, in particular, give the effect of amplitude of the waves on the group velocity. His papers on patterns of ship waves (DlO and Dl1) give two formulae in closed form from which the patterns can be determined simply, be they of gravity waves, gravity-capillary waves, or internal waves.

    The theory is based on three things: (1) The relation between the local wavenumber and the local wave velocity, i.e., the elementary dispersion formula, (2)the principle of stationary phase, and (3) the requirement that the component of the ship velocity in a direction normal to the local wave front be equal to the local wave velocity, in other words, the requirement of steady pattern. From Yih's formulae, all the known results of Kelvin, Havelock, and others for all sorts of waves can be obtained readily, without the rather obscure reasoning initiated by Lord Kelvin. These formulae have brought the theory of ship waves within the reach of fourth-year undergraduates.

    Yih's work on jets, plumes, and diffusion stems from his early work on convective plumes caused by a point source of heat. In this area, he rediscovered the transformations of Zel'dovich (1937) in 1948, while he was a graduate student at the University of Iowa, for laminar convection from a point or line source of heat, and he was able to give two exact solutions for the cases of Prandtl numbers 1 and 2 (D2 and D5). But, much more important for practical applications are his contributions to turbulent jets and plumes, with or without a cross wind. The similarity solutions for straight turbulent jets were well known in the nineteen thirties. Guided by a dimensional analysis, Yih worked with Hunter Rouse in 1947-48 on the experimental determination of the velocity and temperature distributions in a round turbulent plume caused by a point source of heat. But it was not until 1977 that Yih was able to give the eddy viscosity an expression on dimensional arguments and thus to give analytical solutions for both round and plane turbulent plumes for certain turbulent Prandtl numbers, including the number 1.1 for round plumes (D8), which is important because it is near 1. The eddy viscosity was assumed constant in any plane normal to the plume axis, (D9, D1O). Yih was able to apply the same sort of analysis to jets and plumes in a transverse wind. One interesting analytical result is the double-helix structure of jets and plumes in a transverse wind, which one can observe in a chimney plume on a windy day.

    Finally, it should be noted that Yih's contributions to fluid mechanics is not limited to his own work, which forms a significant segment of the literature on the mechanics of fluid flows. As a teacher, he has imparted his ideas to his students who carried them out and published papers under their own names. To appreciate Yih's contributions to fluid mechanics, one should include all these large quantities of results that he caused to appear by virtue of opening new areas of research and suggesting specific problems that needed attention.

    C. C. Lin




    Reminiscence by Friends of Chia-Shun Yih




    Other Aspects of the Author

    I met Chia-Shun in 1934, when we entered Soochow Senior High School. The oldest hall of the school, where we often took examinations, was called the "purple Sun Hall", in honor of Master Zhu Hsi, founder of neo-Confucianism in the Sung Dynasty. We were not sure whether Master Zhu actually lectured there, but the halo of tradition was real and palpable, and we were fortunate that the education we received at the school was worthy of the hallowed site.

    We had excellent teachers. But we were very anxious to learn as much as we could, and supplemented their teaching by studying independently some English and American books that Chia-Shun brought back from Shanghai, which was a short ride from Soochow by train, after one spring vacation. These books, fortunately mostly well chosen, not only gave us additional instruction and information on mathematics, physics, and chemistry, but also formed our habit of reading scientific books in English rather early in our lives. Furthermore, among the books were literary works by English authors, such as Charles Dickens and Jane Austen. Chia-Shun has since become a devotee of Austen, has read nearly all ocher works, and recalls with nostalgia the compact edition, with deep-blue covers, of Pride and Prejudice that he brought back from Shanghai.

    The text books for English used in our school were edited by our own teachers. Looking back, we are impressed with their good judgment and taste in the selection of material for these books. The English authors represented were Shakespeare (Julius Caesar, The Merchant of Venice), Macauley, Dickens (A Tale of Two Cities), Jane Austen (Pride and Prejudice), George Eliot (The Mill on the Floss), among others, and curiously, also Lord Chesterfield (letters to his son). Among the American authors selected were Nathaniel Hawthorn (The Scarlet Letter) and Washington Irving (The Sketch Book). That this sort of education ill prepared us for ordering a meal or hailing a taxi when we came to America after the War was a minor inconvenience quickly overcome by the discipline of necessity; the influence of these masters of letters has remained with us ever since.

    While we were in high school, the storm was gathering in China and in Europe. Japan had already occupied Manchuria since 1931, and had fought with our armed forces in Shanghai in 1932. Full-scale war between China and Japan finally broke out in 1937, soon after our graduation from high school. We managed to take the entrance examination of the National Central University, in Nanking, about two months before its fall. Our university was moved to Chungking far in the hinterland, by a quick and farsighted decision of Dr. Chia-Luen Lo, our president. While we were students in Chungking, war broke out in Europe in 1939,and the Japanese bombed Pearl Harbor in 1941. The Chinese people were no longer fighting alone.

    Since Dr. Lo's wife and daughters were all Barber scholars at the University of Michigan at different times, it was natural that Chia-Shun should become a friend of the entire Lo family. To illustrate how thoroughly Western culture penetrated Chinese intellectuals one generation ahead of us, I shall repeat here a story about Dr. Lo that Chia-Shun likes to tell. Our university had an extensive library, many delicate instruments and various chemicals that were hard to come by, and much cattle (Holstein cows) of the Agriculture College. To move it from Nanking to Chungking it was necessary not only to crate and transport the books, records, instruments, and chemicals, but also to herd the cattle along the Yangtze River for a distance of some 1,500 miles. It took something like two years for the cows to arrive, and they were by then all shaggy and thin. Dr. Lo encountered them one day unexpectedly in the outskirts of Chungking. He told Chia-Shun that the words that immediately came to his mind were "Ah, du armes Kind, was hat man dir getan!" The warm humanity oft he man! Goethe would have been pleased to have his words so applied. Dr. Lo was educated in Princeton and Berlin, and was a poet and a distinguished educator. Chia-Shun was very fond of him.

    Our independent study of physics, chemistry, and mathematics stood us in good stead when we entered the university. We sailed through the first year without much effort. This gave Chia-Shun an opportunity to satisfy his thirst for Western literature. He was introduced to the romantic poets by Professor YuTa-Ying, and became very fond of John Keats' poetry. He also studied German literature with Professor Shang Dzang-Sun, who gave him, after graduation, a compact book of Goethe's "Die Leiden des jungen Werthers." Chia-Shun negotiated through the treacherous terrain of German grammar, and still managed to enjoy the beauty of the work. A few years later, while pursuing his doctorate at the University of Iowa, which demanded of its Ph.D. candidates proficiency in two foreign languages, Chia-Shun made good use of his German. But since English is not foreign in Iowa and Chinese is not foreign to Chia-Shun, he had to learn another Western language. He chose French, and many years later, while Visiting Professor at the University of Paris and the University of Grenoble, was able to lecture in that language. He was on rather friendly terms with the French nasal sounds and the u's and eu's so deadly to most Americans and is fond of quoting Bernard Shaw's witticism that the French don't care what you say, so long so you pronounce it right. His interest in French finally enabled him to read Stendhal, Flaubert, Maupassant, and the part "Du Cote' de chez Swaun" of Proust's "A la Recherche des Temps Perdus."

    At our university Chia-Shun studied, among other things, mathematics and the theory and design of bridges, and I studied mathematics and the theory of airplane structures. Our college years were spent in make-shift classrooms and laboratories, classes at the crack of dawn to avoid air raids, long hours in the dugouts, military training, and an endless stream of exciting or sad news. One wintry day Japanese planes came and bombed out our simple shower hut, and for weeks afterwards some of us had to bathe in the emerald water of the near-by Chia-Ling River, beautiful but cold. Chia-Shun likes to say that the dominant sensation when one jumps into icy water is an immediate headache. "Afterwards, the shiver in the sun seemed almost pleasant by comparison."

    After graduation Chia-Shun worked first in a hydraulics laboratory in Quanshien, and then for the National Bureau of Bridge Design in Kweiyang. I was then working for the Bureau of Airplane Design in Chengtu. At that stage we both intended to be practicing engineers. Then something changed our lives. By nationwide examinations, the ministry of Education chose forty-two scholars to study in the United States. We were among the forty-two, and after the War came to the United States via India. At Calcutta we boarded the American troop ship "General Hase," which took a month crossing the Red Sea, the Mediterranean, and the Atlantic before reaching New York City on December 28,1945. I then went to Caltech, and Chia-Shun, after a very brief stay at Purdue University, went to the University of Iowa to study fluid mechanics with Hunter Rouse and John McNown, with whom he maintains a warm friendship to this day.

    The story of his work after his graduate studies is best told by the papers included in these volumes. But I know enough of his work to say that Chia-Shun is a superb mathematician. Every paper of his contains some elegant mathematics that cuts through the complexity and clarifies the physics at the core. That is the feature of his work that has earned Chia-Shun the respect and admiration of his colleagues.

    Yet Chia-Shun is much more than a mathematician. His human qualities, his generosity, and his sincerity touch every heart that comes in contact with him. His lively conversation at parties and dinners, his love of Nature, literature, and art (especially impressionist paintings), his knowledge of flowers and trees, and his eagerness to be helpful are well-known to his friends.

    Memories are our most certain possessions. Yet we know they are also ephemeral, and will vanish with the inexorable passage of time. So let me indulge a little in my own remembrance of times past. One memorable trip we took in our high school days was a trip to Yi Shin, to see the stalactites and stalagmites in caves, and the underground running stream, in which one could row a boat for several kilometers. We used candles and kerosene lamps, which we could put out to experience a nearly total darkness in which fluorescent insects and mosses shone. In my recollection, that was the first time that we were truly impressed with our biosphere. Our "senior trip" was made on a small steamer along the Fu Chuen River near Hangchow. Fu Chuen! What a poetic name! Fu means "rich" or, in this context, "rich in", and Chuen means, "spring". It was a limpid river bordered by tallish but not menacing mountains, and that night the moon shone bright. Chia-Shun loves that river. The other river he passionately loves is the Chia-Ling, which flows below where our university used to stand. Its emerald waters again and again found expression in the poems of Li Shang-Yin (of the late Tang Dynasty), Chia-Shun's favorite poet. Chia-Shun's love of Nature literature, and art is so essential apart of him that, without it, he would be greatly impoverished.

    Every life has its sorrows. Most lives have some misfortunes. Chia-Shun's is no exception. But on the whole it seems that his is an enchanted life. I should like to quote a poem by Wang Wei () to conclude this reminiscence:

    Paraphrased, it says: "Would that I could shake off the net that binds me to this dusty world, say good-bye to the hustle and bustle, and, swinging my bamboo stick leisurely, return to the Creek Of Peach Blossoms. "

    People through the ages have searched for the legendary land of the Creek of Peach Blossoms I think some of us are living in it.

    Yuan-Cheng Fung
    La Jolla, 1990



    Chia-Shun Yih was the first Chinese fluid dynamicist I met, and naturally I assumed there were many others like him. I have not yet found one! Of all my friends, in any country, he is the most mercurial, the most emotional, and the most generous-spirited. It is unusual to find these qualities in a person with scientific interests, but in Chia-Shun they are combined happily. Given another life-time, one feels he might prefer to be a painter or a poet; but in his present incarnation he is outstanding as an applied mathematician, quick in perception, skilled in execution, and always on the look-out for a nice result which has both elegance and meat.

    I first met Chia-Shun in the 1950's. We are near-contemporaries, and we both had the good fortune to 'grow up' in research during those exciting early post-war years when there seemed to be so much useful basic research in fluid mechanics to be done. Our areas of specialization were not the same, but we had a few common interests, for example in free convection and hydrodynamic stability. He made some short visits to Cambridge, and fell in love with the beautiful old College buildings, the academic atmosphere, and the old-world values, things which he greatly missed in USA. The idea of spending a whole academic year at Cambridge arose naturally between us, and a subsequent successful application to NSF for a visiting fellowship enabled him to live in Cambridge with his family during 1959-60. He was very interested in stratified fluid flow at the time, and this gave him common ground with a number of his Cambridge friends, especially Brooke Benjamin, Alfred Binnie, and Geoffrey Taylor. He was a marvelous visitor, he enjoyed everything, even the weather, and was especially appreciative of opportunities to observe the Cambridge scene, to learn a little of its history, and to take part in the ceremonies of daily College life. He referred later to this as being the happiest year of his life. Speaking as one of the residents, I can testify that we felt the gain was entirely on our side. His capacity for human warmth and affection and his enthusiasm for research in fluid mechanics win him friends everywhere. May they continue to do so.

    George Batchelor
    Cambridge, August 1990



    My advisor spent the academic year 1963-64 as Visiting Professor at M.I.T. and I as a "terminal" research assistant went along as a "visiting student." During that year Chia-Shun Yih gave an Applied Mathematics Colloquium on his now well-known transformation relating inviscid incompressible fluid flows to those of constant density via the mapping . I recall noting that there was something peculiar about this lecture, though it wasn't until the very end that I realized what it was. The speaker was really enjoying himself. He was enthused by his discovery and was not embarrassed to show it. This is one hallmark of Chia-Shun. He is a person of buoyant spirits and good cheer.

    When I was put up for promotion and tenure at Johns Hopkins, Chia-Shun was asked to be a referee. He not only wrote a very positive letter on my behalf, but he sent a copy of it directly to me. This was a time of "cold war" within the department and my receipt of this letter greatly buoyed my spirits. This is another hallmark of Chia-Shun. He performs acts of kindness for his colleagues, acts that are beyond the call of duty.

    Chia-Shun has been given two great honors by the American Physical Society, the Fluid Dynamics Prize and the Otto Laporte Award. On each occasion he was touched by the honor and obviously pleased by the action of his colleagues. One could, again, feel and share his joy.

    Chia-Shun Yih is credited with several major discoveries. Instability driven by viscosity stratification, subharmonic instabilities in modulated viscous flows and the transformation are some. His 1963 invention of long-wave analysis of free-surface instabilities has given birth to, perhaps, a thousand papers of others. But to me what are even more important are Chia-Shun's characteristics as a man. He is a gentleman, and I am proud and fortunate to know him.

    Stephen H. Davis
    Evanston, 1990