One of the most challenging problems of modern engineering is undoubtedly the prediction of hypersonic flows around space vehicles in reentry conditions. Indeed, the difficulties are numerous: first of all, these flows are very difficult to model, since very complex physical and chemical phenomena take place during the reentry phase; secondly, temperature, velocity and enthalpy are very high and densities are very low, making the reentry process very difficult to reproduce in ground-based experiments. The past three decades have seen important efforts in computational fluid dynam ics relying on the use of supercomputers to simulate these very complicated flows. The numerical simulation based on imperfect models and methods which were es sentially designed for transonic and supersonic flows has still a long way to go in order to be able to predict these hypersonic reentry flows very accurately. This situation has motivated very strong international cooperative efforts with, as the most visible consequences, the EuropelUnited States Short Courses on Hy personics, which were held in Paris, in 1987 [1,2], Colorado Springs in 1989 [3], and Aachen in 1990 [3]. The workshop on Hypersonics whose results are presented and analysed in these volumes is also a direct consequence of this international cooperation. This scien tific event was an initiative of P. Perrier, Head of the Theoretical Aerodynamics Department of DASSAULT AVIATION, who played a key role in the identification of the critical problems and the realisation of experiments, within the Hermes R&D program framework.

The second edition of this volume has been extensively revised. A different version of Chap. 7, reflecting recent significant progress in understanding of spatiotempo ral chaos, is now provided. Much new material has been included in the sections dealing with intermittency in birth-death models and noise-induced phase transi tions. A new section on control of chaotic behavior has been added to Chap. 6. The subtitle of the volume has been changed to better reflect its contents. We acknowledge stimulating discussions with H. Haken and E. Scholl and are grateful to our colleagues M. Bar, D. Battogtokh, M. Eiswirth, M. Hildebrand, K. Krischer, and V. Tereshko for their comments and assistance. We thank M. Lubke for her help in producing new figures for this volume. Berlin and Moscow A. s. Mikhailov April 1996 A. Yu. Loskutov Preface to the First Edition This textbook is based on a lecture course in synergetics given at the University of Moscow. In this second of two volumes, we discuss the emergence and properties of complex chaotic patterns in distributed active systems. Such patterns can be produced autonomously by a system, or can result from selective amplification of fluctuations caused by external weak noise."

Computational methods and modelling is of growing importance in fundamental science as well as in applications in industry and in environmental research. In this topical volume the readers find important contributions in the field of turbulent boundary layers, the Tsunami problem, group invariant solution of hydrodynamic equations, non-linear waves, modelling of the problem of evaporation-condensation, the exact solution of discrete models of the Boltzmann equation etc. The book addresses researchers and engineers both in the mechanical sciences and in scientific computing.

Smol'yakov and Tkachenko's book is a very thorough and detailed survey of the response of hot wires and related trans ducers to a fluctuating flow field. Now that the electronic equipment needed for hot-wire anemometry is so easy to make or cheap to buy, transducer response is the most critical part of the subject - except for the fragility of the sensing element , for which textbooks are no remedy! We hope that this book will be useful to all students and research workers concerned with the theory or practice of these devices or the interpretation of results. Peter Bradshaw Imperial College London v Preface "The importance of experimental data and of experimentally established general properties is often underestimated in the study of turbulence . . . *. The most direct path is to use experimentally established properties as the foundation upon which models explaining these properties can be constructed. " M. D. Millionshchikov Turbulence belongs to a class of physical phenomena that are very frequently encountered in both nature and technology. It is the most common and also the most complicated form of motion of real liquids and gases. It is observed in the oceans, in the atmosphere, and in a very wide range of systems in engineering. The rational design of airplanes, rockets, ships, dams, hydroelectric plant, canals, turbines, ventilators, and many other technological systems must involve the consideration of turbulence.

This book goes beyond the scope of other works in the field with its thorough treatment of applications in a wide variety of disciplines. The third edition features a new section on constants of motion and symmetry and a new appendix on the Lorentz-Legendre expansion.

Topology is a relatively young and very important branch of mathematics. It studies properties of objects that are preserved by deformations, twistings, and stretchings, but not tearing. This book deals with the topology of curves and surfaces as well as with the fundamental concepts of homotopy and homology, and does this in a lively and well-motivated way. There is hardly an area of mathematics that does not make use of topological results and concepts. The importance of topological methods for different areas of physics is also beyond doubt. They are used in field theory and general relativity, in the physics of low temperatures, and in modern quantum theory. The book is well suited not only as preparation for students who plan to take a course in algebraic topology but also for advanced undergraduates or beginning graduates interested in finding out what topology is all about. The book has more than 200 problems, many examples, and over 200 illustrations.

This volume consists of edited papers presented at the International Symposion Gas Phase Chemical Reaction Systems: Experiments and Models 100 Years After Max Bodenslein, held at the Internationales Wissenschaftsforum Heidelberg (IWH) in Heidelberg during July 25-28, 1995. The intention of this symposion was to bring together leading researchers from the fields of reaction dynamics, kinetics, catalysis and reactive flow model ling to discuss and review the advances in the understanding of chemical kinetics about 100 years after Max Bodenstein's pioneering work on the "hydrogen iodine reaction", which he carried out at the Chemistry Institute of the University of Heidelberg. The idea to focus in his doctoral thesis [1] on this reaction was brought up by his supervisor Victor Meyer (successor of Robert Bunsen at the Chemistry Institute of the University of Heidelberg) and originated from the non reproducible behaviour found by Bunsen and Roscoe in their early photochemical investigations of the H2/Cl2 system [2] and by van't Hoff [3], and V. Meyer and co-workers [4] in their experiments on the slow combustion of H2/02 mixtures.

Computation of Unsteady Internal Flows provides an in-depth understanding of unsteady flow modeling and algorithms. This understanding enables suitable algorithms and approaches for particular fields of application to be selected. In addition, the understanding of the behavior of algorithms gained allows practitioners to use them more safely in existing codes, enabling meaningful results to be produced more economically. Features of Computation of Unsteady Internal Flows: * Specialized unsteady flow modeling algorithms, their traits, and practical tips relating to their use are presented. * Case studies considering complex, practically significant problems are given. * Source code and set-up files are included. Intended to be of a tutorial nature, these enable the reader to reproduce and extend case studies and to further explore algorithm performances. * Mathematical derivations are used in a fashion that illuminates understanding of the physical implications of different numerical schemes. Physically intuitive mathematical concepts are used. * New material on adaptive time stepping is included.GBP/LISTGBP Audience: Researchers in both the academic and industrial areas who wish to gain in-depth knowledge of unsteady flow modeling will find Computation of Unsteady Internal Flows invaluable. It can also be used as a text in courses centered on computational fluid dynamics.

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.

Cellular automata are fully discrete dynamical systems with dynamical variables defined at the nodes of a lattice and taking values in a finite set. Application of a local transition rule at each lattice site generates the dynamics. The interpretation of systems with a large number of degrees of freedom in terms of lattice gases has received considerable attention recently due to the many applications of this approach, e.g. for simulating fluid flows under nearly realistic conditions, for modeling complex microscopic natural phenomena such as diffusion-reaction or catalysis, and for analysis of pattern-forming systems. The discussion in this book covers aspects of cellular automata theory related to general problems of information theory and statistical physics, lattice gas theory, direct applications, problems arising in the modeling of microscopic physical processes, complex macroscopic behavior (mostly in connection with turbulence), and the design of special-purpose computers.

The inaugural Symposium on Turbulent Shear Flows was held at The Pennsylvania State University in 1977. Thereafter the locations for the biennial symposium have alternated between the USA and Europe. However, the ninth Symposium on Turbu lent Shear Flows was awarded to Japan in recognition of the strong support researchers of the Pacific Rim countries have given previous symposia. The University of Kyoto was the host institution and the meeting was held in the Inter national Conference Hall. The Local Arrangements Committee did a superb job scheduling traditional Japanese dinners and arranging visits to the many cultural treasures in the Kyoto region. The meeting attracted more than 260 offers of papers. Thirty-three sessions were scheduled to accommodate the 138 papers accepted for oral presentation. In addition a poster session was scheduled on each of the three days to accommodate a total of 42 poster presentations. From the presentations at the symposium 24 have been selected for inclusion in this volume. The authors of these papers have revised them taking into consideration comments made during their oral presentation and recommendations made by the Editors. Four subject areas are identified, namely closures and fundamentals, free flows, wall flows, and combustion and recirculating flows. Eminent authorities have prepared introductory articles fot each topic to put the individual contributions in context with each other and with related research.

The main topics reflect the fields of mathematics in which Professor O.A. Ladyzhenskaya obtained her most influential results. One of the main topics considered in the volume is the Navier-Stokes equations. This subject is investigated in many different directions. In particular, the existence and uniqueness results are obtained for the Navier-Stokes equations in spaces of low regularity. A sufficient condition for the regularity of solutions to the evolution Navier-Stokes equations in the three-dimensional case is derived and the stabilization of a solution to the Navier-Stokes equations to the steady-state solution and the realization of stabilization by a feedback boundary control are discussed in detail. Connections between the regularity problem for the Navier-Stokes equations and a backward uniqueness problem for the heat operator are also clarified. Generalizations and modified Navier-Stokes equations modeling various physical phenomena such as the mixture of fluids and isotropic turbulence are also considered. Numerical results for the Navier-Stokes equations, as well as for the porous medium equation and the heat equation, obtained by the diffusion velocity method are illustrated by computer graphs. Some other models describing various processes in continuum mechanics are studied from the mathematical point of view. In particular, a structure theorem for divergence-free vector fields in the plane for a problem arising in a micromagnetics model is proved. The absolute continuity of the spectrum of the elasticity operator appearing in a problem for an isotropic periodic elastic medium with constant shear modulus (the Hill body) is established. Time-discretization problems for generalized Newtonian fluids are discussed, the unique solvability of the initial-value problem for the inelastic homogeneous Boltzmann equation for hard spheres, with a diffusive term representing a random background acceleration is proved and some qualitative properties of the solution are studied. An approach to mathematical statements based on the Maxwell model and illustrated by the Lavrent'ev problem on the wave formation caused by explosion welding is presented. The global existence and uniqueness of a solution to the initial boundary-value problem for the equations arising in the modelling of the tension-driven Marangoni convection and the existence of a minimal global attractor are established. The existence results, regularity properties, and pointwise estimates for solutions to the Cauchy problem for linear and nonlinear Kolmogorov-type operators arising in diffusion theory, probability, and finance, are proved. The existence of minimizers for the energy functional in the Skyrme model for the low-energy interaction of pions which describes elementary particles as spatially localized solutions of nonlinear partial differential equations is also proved. Several papers are devoted to the study of nonlinear elliptic and parabolic operators. Versions of the mean value theorems and Harnack inequalities are studied for the heat equation, and connections with the so-called growth theorems for more general second-order elliptic and parabolic equations in the divergence or nondivergence form are investigated. Additionally, qualitative properties of viscosity solutions of fully nonlinear partial differential inequalities of elliptic and degenerate elliptic type are clarified. Some uniqueness results for identification of quasilinear elliptic and parabolic equations are presented and the existence of smooth solutions of a class of Hessian equations on a compact Riemannian manifold without imposing any curvature restrictions on the manifold is established.

Turbulence is a major problem facing modern societies. It makes airline passengers return to their seats and fasten their seatbelts but it also creates drag on the aircraft that causes it to use more fuel and create more pollution. The same applies to cars, ships and the space shuttle. The mathematical theory of turbulence has been an unsolved problems for 500 years and the development of the statistical theory of the Navier-Stokes equations describes turbulent flow has been an open problem. The Kolmogorov-Obukhov Theory of Turbulence develops a statistical theory of turbulence from the stochastic Navier-Stokes equation and the physical theory, that was proposed by Kolmogorov and Obukhov in 1941. The statistical theory of turbulence shows that the noise in developed turbulence is a general form which can be used to present a mathematical model for the stochastic Navier-Stokes equation. The statistical theory of the stochastic Navier-Stokes equation is developed in a pedagogical manner and shown to imply the Kolmogorov-Obukhov statistical theory. This book looks at a new mathematical theory in turbulence which may lead to many new developments in vorticity and Lagrangian turbulence. But even more importantly it may produce a systematic way of improving direct Navier-Stokes simulations and lead to a major jump in the technology both preventing and utilizing turbulence.

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."

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."

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.

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."

"Hydrothermal and Supercritical Water Processes" presents an overview on the properties and applications of water at elevated temperatures and pressures. It combines fundamentals with production process aspects. Water is an extraordinary substance. At elevated temperatures (and pressures)its properties change dramatically due to the modifications of the molecular structure of bulk water that varies from a stable three-dimensional network, formed by hydrogen bonds at low and moderate temperatures, to an assembly of separated polar water molecules at high and supercritical temperatures. With varying pressure and temperature, water is turned from a solvent for ionic species to a solvent for polar and non-polar substances. This variability and an enhanced reactivity of water have led to many practical applications and to even more research activities, related to such areas as energy transfer, extraction of functional molecules, unique chemical reactions, biomass conversion and fuel materials processing, destruction of dangerous compounds and recycling of useful ones, growth of monolithic crystals, and preparation of metallic nanoparticles.

This book provides an introduction into the wide range of activities that are possible in aqueous mixtures. It is organized to facilitate understanding of the main features, outlines the main applications, and gives access to further information
Summarizes fundamental properties of water for engineering applicationsCompares process and reactor designs Evaluates processes from thermodynamic, economic, and social impact viewpoints"

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."

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."