The present volume entitled "Perspectives in Turbulence Stud ies" is dedicated to Dr. Ing. E. h. Julius C. Rotta in honour of his 75th birthday. J. C. Rotta, born on January 1, 1912, started his outstanding career in an unusual way, namely in a drawing office (1928 - 1931). At the same time he - as a purely self taught perso- took a correspondence course in airplane construction. From 1934 to 1945 he worked in the aircraft industry on different subjects in the fields of flight mechanics, structures, air craft design, and aerodynamics. In 1945 he moved to Gottingen and worked from that time at the Aerodynamische Versuchsanstalt (AVA, now DFVLR) and the Max-Planck-Institut fur Stromungsforschung (1947-1958), interrupted only by a stay in the U. S. at the Glenn L. Martin Company (1954 - 1955) and a visiting professorship at the Laval University in Quebec, Canada (1956). Already during his activities in industry, Dr. Rotta discovered his special liking for aerodynamics. In Gottingen, he was attracted by Ludwig Prandtl's discussions about problems associated with turbulence and in particular his new contribution to fully developed turbulence, published in 1945. At that time, W. Heisenberg and C. F. v. Weizacker pub lished their results on the energy spectra of isotropic turbu lence at large wave numbers. Since that time his main research interest in reasearch has been in turbulence problems."

A detailed look at some of the more modern issues of hydrodynamic stability, including transient growth, eigenvalue spectra, secondary instability. It presents analytical results and numerical simulations, linear and selected nonlinear stability methods. By including classical results as well as recent developments in the field of hydrodynamic stability and transition, the book can be used as a textbook for an introductory, graduate-level course in stability theory or for a special-topics fluids course. It is equally of value as a reference for researchers in the field of hydrodynamic stability theory or with an interest in recent developments in fluid dynamics. Stability theory has seen a rapid development over the past decade, this book includes such new developments as direct numerical simulations of transition to turbulence and linear analysis based on the initial-value problem.

This volume contains a selection of the papers presented at the Eighth Symposium on Turbulent Shear Flows held at the Technical University of Munich, 9-11 September 1991. The first of these biennial international symposia was held at the Pennsylvania State Uni versity, USA, in 1977; subsequent symposia have been held at Imperial College, London, England; the University of California, Davis, USA; the University of Karlsruhe, Ger many; Cornell University, Ithaca, USA; the Paul Sabatier University, Toulouse, France; and Stanford University, California, USA. The purpose of this series of symposia is to provide a forum for the presentation and discussion of new developments in the field of turbulence, especially as related to shear flows of importance in engineering and geo physics. From the 330 extended abstracts submitted for this symposium, 145 papers were presented orally and 60 as posters. Out of these, we have selected twenty-four papers for inclusion in this volume, each of which has been revised and extended in accordance with the editors' recommendations. The following four theme areas were selected after consideration of the quality of the contributions, the importance of the area, and the selection made in earlier volumes: - wall flows, - separated flows, - compressibility effects, - buoyancy, rotation, and curvature effects. As in the past, each section corresponding to the above areas begins with an introduction by an authority in the field that places the individual contributions in context with one another and with related research.

The Third European Turbulence Conference was held at the Royal Institute of Technology, Sweden, from the 3rd to the 6th of July 1990 under the auspices of the European Mechanics Committee. This series of conferences is primarily devoted to fundamental aspects of turbulence and aimed at bringing together engineers, physicists and mathematicians. The scientific committee - serving also as the European Turbulence Con ference Committee - consisted of the following members: G. Comte-Bellot (Lyon), H.H. Fernholz (Berlin), H. Fiedler (Berlin), U. Frisch (Nice), J.c.R. Hunt (Cambridge), J. Jimenez (Madrid), A.V. Johansson (Stockholm), E. Krause (Aachen), G. Ooms (Amsterdam), and 1. Rhyming (Lausanne). The local organizing committee consisted of A.V. Johansson and P.H. Alfredsson. The conference programme comprised six invited lectures, given by D. Bechert, Y. Gagne, J. Jimenez, M.T. Landahl, C.H. Priddin and G.M. Za slavsky, and 96 contributions given as either oral or poster presentations.There were 172 participants from 17 countries. The members of this scientific com mittee acted as referees for the selection and scrutiny of the papers published in these proceedings. The main topics of this meeting were turbulence structure; transition and dynamical systems; turbulent combustion and mixing; turbulence affected by body forces; turbulence modelling; drag reduction and turbulence control; and novel experimental techniques. During the first evening of the conference the new MTL low-speed, low-turbulence wind-tunnel at KTH was inaugurated. The wind-tunnel outline, also acting as decoration for the "official conference cakes," is shown below."

Separated flows and jets are closely linked in a variety of applications. They are of great importance in various fields of fluid mechanics including vehicle efficiency, technical branches concerned with gas/liquid flows, atmospheric effects on various constructions, etc. Knowledge of the physics of separated flows and jets and the development of reliable control techniques are prerequisite for future progress in the field. These aspects were in focus during the IUTAM-Symposium which was held in Novosibirsk, 9-13 July, 1990. This volume contains a selection of papers presenting recent results of theoretical and numerical studies as well as experimental work on separated flows and jets. The topics include sub- and supersonic, laminar and turbulent separation as well as organized structures in separated flows and jets. The reader will find here the state of the art and major trends for research in this field of aero-hydrodynamics.

Since 1964 the main function of the European Mechanics Committee has been to arrange Euromech Colloquia. These are three- or four-day meetings for the discussion of current research on a specified and relatively narrow topic in mechanics, by about 50 specialists chosen for their active involvement in research in that topic. The organization of each Euromech Colloquium is entrusted by the Committee to one or two selected scientists of repute in the field, and these organizers are enjoined to achieve a friendly and informal forum for discussion, with a minimum of paper work and expenditure. Over 220 Euromech Colloquia have been held since 1964 (about 40 each in France, West Germany and Britain and the remainder in 18 countries in both western and eastern Europe) on a wide range of topics drawn from the mechanics of solid materials, hydrodynamics, gas dynamics and mechanical systems. The Committee believes that collectively, Euromech Colloquia have made a significant contribution to the exchange of ideas on topics in mechanics within Europe and have thereby helped to overcome the barriers to easy scientific communication in that sorely divided continent. A few years ago the European Mechanics Committee turned its atten tion to the possible need for European conferences on a larger scale than Euromech Colloquia."

Recently, there have been significant advances in the fields of high-enthalpy hypersonic flows, high-temperature gas physics, and chemistry shock propagation in various media, industrial and medical applications of shock waves, and shock-tube technology. This series contains all the papers and lectures of the 19th International Symposium on Shock Waves held in Marseille in 1993. They are published in four topical volumes, each containing papers on related topics, and preceded by an overview written by a leading international expert. The volumes may be purchased independently.

Both of the authors of this book are disciples and collaborators of the Brussels school of thermodynamics. Their particular domain of competence is the application of numerical methods to the many highly nonlinear problems which have arisen in the context of recent developments in the thermodynamics of irreversi ble processes: stability of states far from equilibrium, search for marginal critical states, bifwrcation phenomena, multiple stationnary states, dissipative structures, etc. These problems cannot in general be handled using only the clas sical and mathematically rigorous methods of the theory of differential, partial differential, and int grodifferential equations. The present authors demonstrate how approximate methods, re lyi ng usually on powerful computers, lead to significant progress in these areas, if one is prepa red to accept a certain lack of rigor, such as, for example, the lack of proof for the convergence of the series used in the context of problems which are not self adjoint, nor even linear. The results thus obtained must consequently be submit ted to an exacting confrontation with experimental observations. - Even though, the '1 imited information obtained concerning the, often unsuspec ted, mechanisms underlying the observed phenomena is both precious and frequently sufficient. This information results from the properties of the trial functions best suited to the constraints of the problem such as the initial, boundary, and "feedback" conditions, and the analysis of their behavior in the course of the evolution of the system."

The General Assembly of the International Union of Theoreti cal and Applied Mechanics decided in Cambridge, United King dom, in 1982 to arrange the Symposium on Optical Methods in the Dynamics of Fluids and Solids. This decision was stimu lated by the fact that optical diagnostic methods are a more and more important tool in experimental mechanics. In contrast to the foregoing Symposium in Poitiers. 1976, which was devoted exclusively to optical methods in the me chanics of solids, it was a fruitful idea to bring together during the present Symposium scientists engaged in optical methods in the dynamics of all phases. It was proposed by the International Scientific Committee of the Symposium that contributions in experimental fluid dynamics should deal with transition from laminar to turbu lent flow, compressible fluid flow including high temperatu re flow, non-equilibrium phenomena in fluid dynamics and the interaction of fluid flow with solid boundaries and bodies. As regard. the mechanics of solids, the contributions should deal with the application of optical methods in the wave propagation in shock loaded bodies, in phenomena connected with the fracture mechanism, in nonstationary Vibrations of elements and parts of systems and in nonstationary strains in structures. The International Scientific Committee preferred to avoid invited lectures and in cooperation with the National Com mittees of IUTAM called for contribution from individual countries.

would like to thank all those people in the life of Walter Noll I who helped me to write this book: Chancellor ofIndiana Univer- sity Professor Herman B. Wells, Dr. Klaus Andre, Professor Bemard D. Coleman, Stella DeVito, Dr. Peter Frankel, Professor Morton E. Gurtin, Studiendirektor i.R. Rudolf Hohensee, Faye Mark, Profes- sor Victor J. Mizel, Professor Dietrich Morgenstern, Professor Ralph Raimi, Professor Juan J. Schaffer, Professor Clifford A. Truesdell III, Professor Epifanio G. Virga, Dr. Paul Winkler and many oth- ers. I will be always indebted to Inge Lind, my teacher and friend, for a careful reading ofthe manuscript and her invaluable comments. This book would never have been written without the compas- sion and assistance of many people in Bochum and Marburg. I am especially grateful to Ursula Arras, Dieter and Karola Behm, Achim Buhl, Birgit Berger, Professor Klaus Bohmer, Ute Hagen, Irene Jo- raszik, Professor Hans-Heinrich Korle, Elke and Manfred Kuhne, Jutta Kuster, Johannes Lind, Inka Lins, Annemarie Matt, Astrid Milenz, Matthias Naher, Eckart Rosch, Manfred Schonsee, Profes- sor Werner von Seelen, Christa Seip, Heike Willig, Anita Wolf and many others.

WAVE TURBULENCE is a state of a system of many simultaneously excited and interacting waves characterized by an energy distribution which is not in any sense close to thermodynamic equilibrium. Such situations in a choppy sea, in a hot plasma, in dielectrics under arise, for example, a powerful laser beam, in magnets placed in a strong microwave field, etc. Among the great variety of physical situations in which wave turbulence arises, it is possible to select two large limiting groups which allow a detailed analysis. The first is fully developed wave turbulence arising when energy pumping and dissipation have essentially different space scales. In this case there is a wide power spectrum of turbulence. This type of turbulence is described in detail e. g. in Zakharov et al. 1 In the second limiting case the scales in which energy pumping and dissipation occur are the same. As a rule, in this case a narrow, almost singular spectrum of turbulence appears which is concentrated near surfaces, curves or even points in k-space. One of the most important, widely investigated and instructive examples of this kind of turbulence is parametric wave turbulence appearing as a result of the evolution of a parametric instability of waves in media under strong external periodic modulation (laser beam, microwave electromagnetic field, etc. ). The present book deals with parametric wave turbulence.

This IMA Volume in Mathematics and its Applications TWO PHASE FLOWS AND WAVES is based on the proceedings of a workshop which was an integral part of the 1988-89 IMA program on NONLINEAR WAVES. The workshop focussed on the development of waves in flowing composites. We thank the Coordinating Commit tee: James Glimm, Daniel Joseph, Barbara Keyfitz, Andrew Majda, Alan Newell, Peter Olver, David Sattinger and David Schaeffer for planning and implementing the stimulating year-long program. We especially thank the Workshop Organizers, Daniel D. Joseph and David G. Schaeffer for their efforts in bringing together many of the major figures in those research fields in which modelling of granular flows and suspensions is used. Avner Friedman Willard Miller, Jr. PREFACE This Workshop, held from January 3-10,1989 at IMA, focused on the properties of materials which consist of many small solid particles or grains. Let us distinguish the terms granular material and suspension. In the former, the material consists exclusively of solid particles interacting through direct contact with one another, either sustained frictional contacts in the case of slow shearing or collisions in the case of rapid shearing. In suspensions, also called two phase flow, the grains interact with one another primarily through the influence of a viscous fluid which occupies the interstitial space and participates in the flow. (As shown by the lecture of I. Vardoulakis (not included in this volume), the distinction between these two idealized cases is not always clear."

Fundamentals of Continuum Mechanics of Soils provides a long-needed general scheme for the study of the important yet problematic material of soil. It closes the gap between two disciplines, soil mechanics and con- tinuum mechanics, showing that the familiar concepts of soil mechanics evolve directly from continuum mechanics. It confirms concepts such as pore pressures, cohesion and dependence of the shear stress on consolidation, and rejects the view that continuum mechanics cannot be applied to a material such as soil. The general concepts of continuum mechanics, field equations and constitutive equations are discussed. It is shown how the theory of mixtures evolves from these equations and how, along with energetics and irrevers- ible thermodynamics, it can be applied to soils. The discussion also sheds light on some aspects of mechanics of materials, especially compressible materials. Examples are the introduction of the Hencky measure of strain, the requirement of dual constitutive equations, and the dependence of the spent internal energy on the stored internal energy. Researchers in engineering mechanics and material sciences may find that the results of experiments on soils can be generalized and extended to other materials. The book is a reference text for students familiar with the fundamentals of mechanics, for scholars of soil engineering, and for soil scientists. It is also suitable as an advanced undergraduate course in soil mechanics.

This is the second of four volumes on the Navier-Stokes equations, specifically on Nonlinear Stationary Problems. The volumes deal with the fundamental mathematical properties of the Navier-Stokes equations, such as existence, regularity and uniqueness of solutions, and, for unbounded domains, their asymptotic behavior. The work is an up-to-date and detailed investigation of these problems for motions in domains of different types: bounded, exterior and domain with noncompact boundaries. Throughout the work, main problems which, so far, remain open are pointed out and for some of these conjectures are offered. New results are presented throughout, while several classical subjects are treated in a completely original way. The work is mathematically self contained, requiring no specific background. The 200-plus exercises along with the chapter summaries and questions make this an excellent textbook for any theoretical Fluid Mechanics course; it is suitable as well for self teaching. It is set up to remain useful as a reference or dictionary.

The ocean has entranced mankind for as long as we have gazed upon it, traversed it, dived into it, and studied it. It remains ever changing and seemingly never changing. Each wave that progresses through the. imme diate surf zone on every coast is strikingly different, yet the waves come again and again, as if never to end. The seasons come with essential reg ularity, and. yet each is individual-whatever did happen to that year of the normal rainfall or tidal behavior? This fascination with the currents of the ocean has always had a most immediate practical aspect: shipping, transportation, commerce, and war have depended upon our knowledge, when we had it, and floundered on our surprising ignorance more often than we wish to reflect. These important practical issues have commanded attention from commercial, academic, and military research scientists and engineers from the earliest era of organized scientific investigation. The matter of direct and insistent investigation was from the outset the behavior of ocean currents with long time scales; namely, those varying on annual or at least seasonal cycles. Planning for all the named enterprises depended, as they still do, of course, on the ability to predict with some certainty this class of phenomena. That ability, as with most physical sci ence, is predicated on a firm basis of observational fact to establish what, amorig the myriad of mathematical possibilities, is chosen by Nature as her expression of fact."

This volume contains the proceedings of the ICASE/LaRC Work- shop on the "Algorithmic Trends for Computational Fluid Dynamics (CFD) in the 90's" conducted by the Institute for Computer Applica- tions in Science and Engineering (ICASE) and the Fluid Mechanics Division of NASA Langley Research Center during September 15-17, 1991. The purpose of the workshop was to bring together numerical analysts and computational fluid dynamicists i) to assess the state of the art in the areas of numerical analysis particularly relevant to CFD, ii) to identify promising new developments in various areas of numerical analysis that will have impact on CFD, and iii) to establish a long-term perspective focusing on opportunities and needs. This volume consists of five chapters - i) Overviews, ii) Accelera- tion Techniques, iii) Spectral and Higher-Order Methods, iv) Multi- Resolution/ Subcell Resolution Schemes (including adaptive meth- ods), and v) Inherently Multidimensional Schemes. Each chapter covers a session of the Workshop. The chapter on overviews contains the articles by J. L. Steger, H.-O. Kreiss, R. W. MacCormack, O.

The main aim of this paper is to present some new and general results, ap plicable to the the equations of two phase flow, as formulated in geothermal reservoir engineering. Two phase regions are important in many geothermal reservoirs, especially at depths of order several hundred metres, where ris ing, essentially isothermal single phase liquid first begins to boil. The fluid then continues to rise, with its temperature and pressure closely following the saturation (boiling) curve appropriate to the fluid composition. Perhaps the two most interesting theoretical aspects of the (idealised) two phase flow equations in geothermal reservoir engineering are that firstly, only one component (water) is involved; and secondly, that the densities of the two phases are so different. This has led to the approximation of ignoring capillary pressure. The main aim of this paper is to analyse some of the consequences of this assumption, especially in relation to saturation changes within a uniform porous medium. A general analytic treatment of three dimensional flow is considered. Pre viously, three dimensional modelling in geothermal reservoirs have relied on numerical simulators. In contrast, most of the past analytic work has been restricted to one dimensional examples."

Featuring a foreword by the astronaut Ulf Merbold, this book is devoted to interfaces between two fluids, that is, between a liquid and a gas or between two liquids. It is the first review on the subject, providing an up-to-date overview.

Moving Loads on Ice Plates is a unique study into the effect of vehicles and aircraft travelling across floating ice sheets. It synthesizes in a single volume, with a coherent theme and nomenclature, the diverse literature on the topic, hitherto available only as research journal articles. Chapters on the nature of fresh water ice and sea ice, and on applied continuum mechanics are included, as is a chapter on the subject's venerable history in related areas of engineering and science. The most recent theories and data are discussed in great depth, demonstrating the advanced state of the modelling and experimental field programmes that have taken place. Finally, results are interpreted in the context of engineering questions faced by agencies operating in the polar and subpolar regions. Although the book necessarily contains some graduate level applied mathematics, it is written to allow engineers, physicists and mathematicians to extract the information they need without becoming preoccupied with details. Structural, environmental, civil, and offshore engineers, and groups who support these industries, particularly within the Arctic and Antarctic, will find the book timely and relevant.

The present volume, with the exception of the introductory chapter, consists of papers delivered at the workshop entitled "The Impact of Supercomputers on the Next Decade of Computational Fluid Dynamics," The workshop, which took place in Jerusalem, Israel during the week of December 16, 1984, was initiated by the National Science Foundation of the USA (NSF), by the Ministry of Science and Development, Israel (IMSD), and co-sponsored by the National Aeronautics and Space Administration (NASA), the Office of Scientific Research of the U.S. Air Force (AFOSR), Tel Aviv University and Massachusetts Institute of Technology. The introductory chapter attempts to summarize what transpired at the workshop. The genesis of the workshop was an agreement between NSF and Il1S, signed in the spring of 1983, to conduct a series of bi-national work shops and symposia. This workshop represented the first activity spon sored under the agreement. The undersigned were selected by their respective national bodies to act as co-coordinators and organizers of the workshop. The first question that we faced was to decide upon a topic. In the past few years the field of CFD has mushroomed and consequently there have been many meetings, symposia, workshops, congresses, etc."

This collection of articles has its origin in a meeting which took place June 12-15, 1989, on the grounds of Salve Regina College in Newport, Rhode Island. The meeting was blessed by beautiful, balmy weather and an idyllic setting. The sessions themselves took place in Ochre Court, one of the elegant and stately old summer cottages for which Newport is acclaimed. Lectures were presented in the grand ballroom overlooking the famous Cliff Walk and Block Island Sound. Counter to general belief, the pleasant surroundings did not appear to encourage truancy or in any other way diminish the quality of the meeting. On the contrary, for the four days of the meeting there was a high level of excitement and optimism about the new perspectives in turbulence, a tone that carried over to lively dinner and evening discussions. The participants represented a broad range of backgrounds, extending from pure mathemat ics to experimental engineering. A dialogue began with the first speakers which cut across the boundaries and gave to the meeting a mood of unity which persisted."

Research on laminar flow and its transition to turbulent flow has been an important part of fluid dynamics research during the last sixty years. Since transition impacts, in some way, every aspect of aircraft performance, this emphasis is not only understandable but should continue well into the future. The delay of transition through the use of a favorable pressure gradient by proper body shaping (natural laminar flow) or the use of a small amount of suction (laminar flow control) was recognized even in the early 1930s and rapidly became the foundation of much of the laminar flow research in the U.S. and abroad. As one would expect, there have been many approaches, both theoretical and experimental, employed to achieve the substantial progress made to date. Boundary layer stability theories have been formu lated and calibrated by a good deal of wind tunnel and flight experiments. New laminar now airfoils and wings have been designed and many have been employed in aircraft designs. While the early research was, of necessity, concerned with the design of subsonic aircraft interest has steadily moved to higher speeds including those appropriate to planetary entry. Clearly, there have been substantial advances in our understanding of transition physics and in the development and application of transition prediction methodolo gies to the design of aircraft."

Dealing with aerodynamics in the broadest sense, this book discusses, in addition to aeroplanes, the aerodynamics of cars and birds, and the motion of diverse objects through air and water. The fundamental notions of mechanics and fluid dynamics are clearly explained, while the underlying science is discussed rigorously, but using only elementary mathematics, and then only occasionally. To put the science into its human context, the author describes -- with many illustrations -- the history of human attempts to fly and discusses the social impact of commercial aviation as well as the outlook for future developments. This new edition has been brought up to date throughout; solutions to selected exercises have been added, as have new problems and other study aids.

This volume contains the paper presented at the 13th DGLRlST AB- Symposium held at the Technische Universitat Miinchen, November 12 to 14, 2002. STAB is the German Aerospace Aerodynamics Association, founded towards the end of the 70's, whereas DGLR is the German Society for Aeronautics and Astronautics (Deutsche Gesellschaft fUr Luft- und Raumfahrt - Lilienthal Oberth e.V.). The mission of STAB is to foster development and acceptance of the dis- cipline "Aerodynamics" in Germany. One of its general guidelines is to concentrate resources and know-how in the involved institutions and to avoid duplication in research work as much as possible. Nowadays, this is more necessary than ever. The experience made in the past makes it easier now, to obtain new knowledge for solving today's and tomorrow's prob- lems. STAB unites German scientists and engineers from universities, research- establishments and industry doing research and project work in numerical and experimental fluid mechanics and aerodynamics for aerospace and other applications. This has always been the basis of numerous common research activities sponsored by different funding agencies.

For more than ten years we have been working with the ideal linear MHD equations used to study the stability of thermonuc1ear plasmas. Even though the equations are simple and the problem is mathematically well formulated, the numerical problems were much harder to solve than anticipated. Already in the one-dimensional cylindrical case, what we called "spectral pollution" appeared. We were able to eliminate it by our "ecological solution." This solution was applied to the two-dimensional axisymmetric toroidal geometry. Even though the spectrum was unpolluted the precision was not good enough. Too many mesh points were necessary to obtain the demanded precision. Our solution was what we called the "finite hybrid elements." These elements are efficient and cheap. They have also proved their power when applied to calculating equilibrium solutions and will certainly penetrate into other domains in physics and engineering. During all these years, many colleagues have contributed to the construc tion, testing and using of our stability code ERATO. We would like to thank them here. Some ofthem gave partial contributions to the book. Among them we mention Dr. Kurt Appert, Marie-Christine Festeau-Barrioz, Roberto Iacono, Marie-Alix Secretan, Sandro Semenzato, Dr. Jan Vac1avik, Laurent Villard and Peter Merkel who kindly agreed to write Chap. 6. Special thanks go to Hans Saurenmann who drew most of the figures, to Dr."