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.

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.

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."

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."

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 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.

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.

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 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."

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."

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."

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.

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."

This textbook treats Hydro- and Fluid Dynamics, the engineering science dealing with forces and energies generated by fluids in motion, playing a vital role in everyday life. Practical examples include the flow motion in the kitchen sink, the exhaust fan above the stove, and the air conditioning system in our home. When driving a car, the air flow around the vehicle body induces some drag which increases with the square of the car speed and contributes to excess fuel consumption. Engineering applications encompass fluid transport in pipes and canals, energy generation, environmental processes and transportation (cars, ships, aircrafts). This book deals with the topic of applied hydrodynamics. The lecture material is grouped into two complementary sections: ideal fluid flow and real fluid flow. The former deals with two- and possibly three-dimensional fluid motions that are not subject to boundary friction effects, while the latter considers the flow regions affected by boundary friction and turbulent shear. The lecture material is designed as an intermediate course in fluid dynamics for senior undergraduate and postgraduate students in Civil, Environmental, Hydraulic and Mechanical Engineering. It is supported by notes, applications, remarks and discussions in each chapter. Moreover a series of appendices is added, while some major homework assignments are developed at the end of the book, before the bibliographic references.

In the last 20 years the study of nonlinear nonequilibrium phenomena in spa tially extended systems, with particular emphasis on pattern-forming phenomena, has been one of the very active areas in physics, exhibiting interesting ramifi cations into other sciences. During this time the study of the "classic" systems, like Rayleigh-Benard convection and Taylor vortex flow in simple fluids, has also been supplemented by the study of more complex systems. Here liquid crystals have played, and are still playing, a major role. One might say that liquid crystals provide just the right amount and right kind of complexity. They are full of non linearities and give rise to new symmetry classes, which are sometimes actually simpler to deal with qualitatively, but they still allow a quantitative description of experiments in many cases. In fact one of the attractions of the field is the close contact between experimentalists and theorists. Hydrodynamic instabilities in liquid crystals had already experienced a period of intense study in the late 1960s and early 1970s, but at that time neither the ex perimental and theoretical tools nor the concepts had been developed sufficiently far to address the questions that have since been found to be of particular interest. The renewed interest is also evidenced by the fact that a new series of workshops has evolved. The first one took place in 1989 in Bayreuth and united participants from almost all groups working in pattern formation in liquid crystals."

This IUTAM Symposium was the first international conference on asymptotic methods for turbulent shear flows. It was the aim of this Symposium to bring together the experts and research workers to discuss recent work in this field. There was general consensus among the participants of the Symposium, that the asymptotic methods provide powerful tool for turbulence modelling, which ought to be used more intensively in practice in addition to the numerical meth- ods. This was the Scientific Committee: K. Gersten (Germany, Chairman) A. Kluwick (Austria) J. - P. Guiraud (France) F. T. Smith (United Kingdom) V. V. Sychev (Russia) S. Kida (Japan) H. K. Moffat (United Kingdom) J. D. A. Walker (USA) We are very thankful that the Symposium was sponsored by the following organizations: * International Union of Theoretical and Applied Mechanics * Deutsche Forschungsgemeinschaft, Bonn * Gesellschaft der Freunde der Ruhr-Universitiit, Bochum * Institut fur Energie-, System-, Material- und Umwelttechnik e. V. , Bochum * Ruhrgas AG, Essen * Dresdner Bank, Bochum * Kluwer Academic Publishers, Dordrecht * Vieweg-Verlag, Wiesbaden We thank in particular the Rektor of the Ruhr University, Professor M. Bormann, who was host of the Symposium and made possible that the Symposium could take place on the campus. The following persons, who helped in organizing the Symposium and made sure that everything was working smoothly and efficiently during the Symposium, de- serve our special thanks: Bernard Rocklage, Gerta Marliani, Petra Berkner and Th.

We are pleased to present the Proceedings of the Second International Conference on Computational Fluid Dynamics held at the University of Sydney, Australia, from July 15 to 19, 2002. The conference was a productive meeting of scientists, mathematicians and engineers involved in the computation of fluid flow. Keynote lectures were presented in the areas of optimisation, algorithms, turbulence and bio-fluid mechanics. Two hundred and fifty abstracts from many countries were received for con sideration. The executive committee, consisting of A. Lerat, M. Napolitano, J.J. Chattot, N. Satofuka and myself, were responsible for the selection of papers. Each of the members had a separate subcommittee to carry out the evaluation. One hundred and seventy papers were selected of which one hundred and fifty two were presented at the conference. All papers that appear in the proceedings have been peer reviewed by a panel of experts (with a minimum of two for every paper) before publication. The conference was attended by 160 delegates with a minimum of late with drawals. The informal and friendly atmosphere provided by the university sur roundings was highly appreciated, and the technical aspects of the conference were stimulating. It is appropriate here to thank Alain Lerat, the retiring secretary of the international scientific committee of the conference. We also wish to welcome J. J. Chattot who is the incoming secretary."

Most fluid flows of practical importance are fully three-dimensional, so the non-linear instability properties of three-dimensional flows are of particular interest. In some cases the three-dimensionality may have been caused by a finite amplitude disturbance whilst, more usually, the unperturbed state is three-dimensional. Practical applications where transition is thought to be associated with non-linearity in a three- dimensional flow arise, for example, in aerodynamics (swept wings, engine nacelles, etc.), turbines and aortic blood flow. Here inviscid `cross-flow' disturbances as well as Tollmien-Schlichting and Goertler vortices can all occur simultaneously and their mutual non-linear behaviour must be understood if transition is to be predicted. The non-linear interactions are so complex that usually fully numerical or combined asymptotic/numerical methods must be used. Moreover, in view of the complexity of the instability processes, there is also a growing need for detailed and accurate experimental information. Carefully conducted tests allow us to identify those elements of a particular problem which are dominant. This assists in both the formulation of a relevant theoretical problem and the subsequent physical validation of predictions. It should be noted that the demands made upon the skills of the experimentalist are high and that the tests can be extremely sophisticated - often making use of the latest developments in flow diagnostic techniques, automated high speed data gathering, data analysis, fast processing and presentation.

IJ:1 June of 1987 the Center for Applied Mathematics and Computer Science at San Jose State University received a bequest of over half a million dollars from the estate of Mrs. Marie Woodward. In the opening article of this collection of papers Jane Day, the founder of the Center, describes the background that led to this gift. In recognition of the bequest it was decided that a series of Woodward Conferences be established. The First Woodward Conference took place at San Jose State University on June 2-3 1988. The themes of the conference were the Theoretical, Computational and Practical Aspects of Wave Phenomena and these same themes have been used to divide the contributions to this volume. Part I is concerned with papers on theoretical aspects. This section includes papers on pseudo-differential operator techniques, inverse problems and the mathematical foundations of wave propagation in random media. Part II consists of papers that involve significant amounts of computation. Included are papers on the Fast Hartley Transform, computational algorithms for electromagnetic scattering problems, and nonlinear wave interaction problems in fluid mechanics. vi Part III contains papers with a genuine physics flavor. This final section illustrates the widespread importance of wave phenomena in physics. Among the phenomena considered are waves in the atmosphere, viscous fingering in liquid crystals, solitons and wave localization.

The aerodynamics of aircraft at high angles of attack is a subject which is being pursued diligently, because the modern agile fighter aircraft and many of the current generation of missiles must perform well at very high incidence, near and beyond stall. However, a comprehensive presentation of the methods and results applicable to the studies of the complex aerodynamics at high angle of attack has not been covered in monographs or textbooks. This book is not the usual textbook in that it goes beyond just presenting the basic theoretical and experimental know-how, since it contains reference material to practical calculation methods and technical and experimental results which can be useful to the practicing aerospace engineers and scientists. It can certainly be used as a text and reference book for graduate courses on subjects related to high angles of attack aerodynamics and for topics related to three-dimensional separation in viscous flow courses. In addition, the book is addressed to the aerodynamicist interested in a comprehensive reference to methods of analysis and computations of high angle of attack flow phenomena and is written for the aerospace scientist and engineer who is familiar with the basic concepts of viscous and inviscid flows and with computational methods used in fluid dynamics.

This book deals with the application of spectral methods to problems of uncertainty propagation and quanti?cation in model-based computations. It speci?cally focuses on computational and algorithmic features of these methods which are most useful in dealing with models based on partial differential equations, with special att- tion to models arising in simulations of ?uid ?ows. Implementations are illustrated through applications to elementary problems, as well as more elaborate examples selected from the authors' interests in incompressible vortex-dominated ?ows and compressible ?ows at low Mach numbers. Spectral stochastic methods are probabilistic in nature, and are consequently rooted in the rich mathematical foundation associated with probability and measure spaces. Despite the authors' fascination with this foundation, the discussion only - ludes to those theoretical aspects needed to set the stage for subsequent applications. The book is authored by practitioners, and is primarily intended for researchers or graduate students in computational mathematics, physics, or ?uid dynamics. The book assumes familiarity with elementary methods for the numerical solution of time-dependent, partial differential equations; prior experience with spectral me- ods is naturally helpful though not essential. Full appreciation of elaborate examples in computational ?uid dynamics (CFD) would require familiarity with key, and in some cases delicate, features of the associated numerical methods. Besides these shortcomings, our aim is to treat algorithmic and computational aspects of spectral stochastic methods with details suf?cient to address and reconstruct all but those highly elaborate examples.

Substantial progress has been made in the field of fluid mechanics under compensated gravity effects (microgravity). The main task of this disciplinehas evolved tremendously. Starting out with the aim of providing assistance in describing flow problems in other microgravity sciences, microgravityfluid mechanics has itself now become acknowledge as a powerful means of research. The IUTAM Symposium on Microgravity Fluid Mechanics has pro- vided the long-awaited forum for scientists from 15 coun- tries to discuss and concretize the "state-of-the-art" in this discipline. The main themes treated are: Interface Phe- nomena, Convective Processes; Marangoni effects, Solidifica- tion, Combustion, Physico-Chemical Processes, Multiphase Phenomena, Residual Acceleration effects, Fluid Handling and Non-Newtonian Flows.

Sect 2. 317 tinuity surfaces 1. This suggests that a wake pressure Pw be associated with each flow past a bluff body, and that a wake parameter (2. 4) which plays the same role as the cavitation parameter (2. 1), be defined for the flow. This idea has been made the basis of a modified wake theory (ef. Sect. 11) which proves to be in good qu- titative agreement with pressure and drag measurements. It should be emphasized, however, that un h like the cavitation number, the wake parameter is a quantity which is not known a priori, and must be empirically determined in each case. (3) Jet flows. The problem of jet efflux from an orifice is one of the oldest in hydrodynamics and the first to be treated by Fig. 3a. the HELMHOLTZ free streamline theory. Of particular importance for engineering applications is the discharge coefficient Cd' which is defined in terms of the discharge Q per unit time, the pressure P, and the cross-sectional area A of the orifice, by the formula, (2. 5) where e is the fluid density. Two methods of measuring Cd have been most fre quently adopted. In the first the liquid issues from an orifice in a large vessel under the influence of gravity _, -____________ ., (Fig. 3 a), while in the second it 1 L is forced out of a nozzle or pipe under high pressure (Fig. 3 b)."