The first International Symposium on Unsteady Aerodynamics and Aero elasticity of Turbomachines was held in Paris in 1976, and was followed by symposia at Lausanne in 1980, Cambridge in 1984, Aachen in 1987, Bei jing in 1989, and Notre Dame in 1991. The proceedings published following these symposia have become recognized both as basic reference texts in the subject area and as useful guides to progress in the field. It is hoped that this volume, which represents the proceedings of the Sixth International Symposium on Unsteady Aerodynamics of Turbomachines, will continue that tradition. Interest in the unsteady aerodynamics, aeroacoustics, and aeroelasticity of turbomachines has been growing rapidly since the Paris symposium. This expanded interest is reflected by a significant increase in the numbers of contributed papers and symposium participants. The timeliness of the topics has always been an essential objective of these symposia. Another important objective is to promote an international exchange between scien tists and engineers from universities, government agencies, and industry on the fascinating phenomena of unsteady turbomachine flows and how they affect the aeroelastic stability of the blading system and cause the radiation of unwanted noise. This exchange acts as a catalyst for the development of new analytical and numerical models along with carefully designed ex periments to help understand the behavior of such systems and to develop predictive tools for engineering applications.

The present volume entitled "Recent Contributions to Fluid Mechanics" is dedicated to Professor Dr.-Ing. Alfred Walz in honour of his 75th birthday. Alfred Walz, born on 11 May 1907, began his outstanding career as an electrical engineer. A few years after obtaining his university degree he became extremely engaged in fluid dynamics. Walking in the footsteps of Prandtl he was able to direct the development of theoretical activities in an inimitable way. He had the great opportunity to work both as an engaged fluid dynamicist -always trying to get to the bottom of things -and as a popular and patient teacher. To all of these things - in his own words - he gave his heart. Consequently, it is a great pleasure to publish the following 34 contributions summarizing the efforts of 56 authors. These artic les in total cover the wide range of experimental as well as theore tical fluid dynamics and reflect the present state of the art. Moreover, all colleagues and friends of Alfred Walz wish that he may be able to continue his work and his influence on the work of all of us via his enlightening ideas. Friedrichshafen, August 1982 Werner Haase Chairman of the Scientific Committee Table of Contents SURVEY PAPER Shear Layer Studies - Past, Present, Future P. Bradshaw .......................................... ."

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."

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 study has two declared aims: it presents the theoretical basis for a provably ideal comparative process for relaxing flows (ICP) and jus tifies its application to jet and, in particular, rocket engines. This will be treated in two parts. Part I offers a status quo report on current calculation methods, and compiles and explains briefly the most important data on selected pro minent rocket engines. Starting from the phenomenology of the dynamical and physico-chemical conversion processes in the fuel-oxidizer fluid mixture and in the burned gases, the ideal thermodynamic comparative process is then derived - as a defined sequential change of states in the system. In order to render this comparative process readily under standable, it is first applied to an appropriate model gas using alge braic equations for all relevant parameters. This model gas undergoes energy conversion processes without forfeiting the simplicity of pre sentation typical of classical gas dynamics. Above all, examination of this model offers proof that it is generally impermissible to use, as is done in practice, the familiar isentropic equation for flow changes of state continuously propagated in flow tube theory. Elementary calculations immediately indicate essential attributes which are also typical for relaxing, multicomponent, one-phase systems, such as the significant 'pressure drop phenomenon' or the establishment of the steady mass flow rate as an 'eigenvalue' of the comparative pro cess. Their relevance to the RE theory is stressed."

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.

This volume includes revised and extended versions of selected papers presented at the Tenth International Symposium on Applications of Laser Techniques to Fluid Mechanics held at the Calouste Gulbenkian Foundation in Lisbon, during the period of July 10 to 13, 2000. The papers describe instrumentation developments for Velocity, Scalar and Multi-Phase Flows and results of measurements of Turbulent Flows, and Combustion and Engines. The papers demonstrate the continuing and healthy interest in the development of understanding of new methodologies and implementation in terms of new instrumentation. The prime objective of the Tenth Symposium was to provide a forum for the presentation of the most advanced research on laser techniques for flow measurements, and communicate significant results to fluid mechanics. The application of laser techniques to scientific and engineering fluid flow research was emphasized, but contributions to the theory and practice of laser methods were also considered where they facilitate new improved fluid mechanic research. Attention was placed on laser-Doppler anemometry, particle sizing and other methods for the measurement of velocity and scalars, such as particle image velocimetry and laser induced fluorescence.

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."

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.

This volume contains eighteen contributions of work, conducted since 2000 in the French - German Research Programme "Numerical Flow Simulation," which was initiated in 1996 by the Centre National de la Recherche Scientifique (CNRS) and the Deutsche Forschungsgemeinschaft (DFG). The main purpose of this third publication on the research programme is again to give an overview over recent progress, and to make the obtained results available to the public. The reports are grouped, like those in the first and the second publi cation (NNFM 66, 1998, and NNFM 75, 2001), under the four headings "Devel opment of Solution Techniques," "Crystal Growth and Melts," "Flows of React ing Gases, Sound Generation" and "Turbulent Flows." All contributions to this publication were reviewed by a board consisting of T. Alziary de Roquefort (Poi tiers, France), H. W. Buggisch (Karlsruhe, Germany), S. Candel (Paris, France), U. Ehrenstein (Nice, France), Th. Gallouet (Marseille, France), W. Kordulla (Gottingen, Germany), A. Lerat (Paris, France), 1. Piquet (Nantes, France), R. Rannacher (Heidelberg, Germany), G. Warnecke (Magdeburg, Germany), and the editor. The responsibility for the contents of the reports nevertheless lies with the contributors."

Critical regimes of two-phase flows with a polydisperse solid phase form the basis of such widespread industrial processes as separation of various powdery materials and minerals dressing. It is impossible to describe such complicated flows analytically. Therefore, this study concentrates on invariants experimentally revealed and theoretically grounded for such flows.

This approach can be compared with the situation in gases, where in order to determine principal parameters of their state, one does not need to measure the kinetic energy and velocity of each molecule and find its contribution to the temperature and pressure. These parameters are determined in a simple way for the system on the whole.

A novel conception of two-phase flows allowing the formulation of their statistical parameters is physically substantiated. On the basis of the invariants and these parameters, a comprehensive method of estimating and predicting mass transfer in such flows is developed.

It is noteworthy that the presented results are mostly phenomenological. Such an approach can be successfully extended to the separation of liquids, gases and isotopes.

The book is intended for students and specialists engaged in chemical technology, mineral dressing, ceramics, microelectronics, pharmacology, power generation, thermal engineering and other fields in which flows carrying solid particles are used in the technological process.

In the last decades, a lot of effort has been directed towards manipulation of turbulent boundary layers by passive devices such as external manipulators (thin flat plates or aerofoil section devices embedded in the outer layer) and/or internal manipulators (small streamwise grooves acting directly on the inner region) for the purpose of reducing viscous drag. The former are commonly referred to as LEI3U s or BLADEs and the laHer riblets or grooves. Though the details of the mechanisms are not firmly understood, world-wide experimenta.! data are available and consistent enough in order to assert the potential of such devices for turbulent drag reduction. It should be noted that following on from recent and successful flight tests, the concept of using grooved surfaces is rather close to finding industrial applications. During the last few years, in Europe, there has been considerable interest in lookillg at the behaviour of such passi,'e turbulence manipulators. A lot of intense research, concerning both experimental and theoretical studies. has been carried out in some European research centres. For the last fi\'e years. informal gatherings. called ,.\ \'orking Pi\l'ty i\Ieetings" , have been set up, once a year; the aim of these meetings is not only to bring together European researchers acti,'e in the field of turbulent drag reduction by passi"e means and to hear about recent de\'c!opments but also to o u t1 ine sui tit ble directions for future research or collaborative programmes.

The field of shock compression science has a long and rich history involving contributions of mathematicians, physicists and engineers over approximately two hundred years. The middle of the nineteenth century was an especially ac tive period with the contributions of Riemann, Rankine and Hugoniot, among others. The middle of the twentieth century saw another increase in activity re lated to shock compression of condensed matter as a result of military applica tions. It was also recognized that shock compression provided a means of sub jecting solids and liquids to extreme states of temperature and pressure difficult to achieve by static means. It has thus become an academic study in its own right. The principal modem contributions to this science were summarized in the landmark paper by Rice, McQueen and Walsh [Solid State Physics, Vol. 6, pp. 1-63, 1958]. As this field has continued to mature, interest has increased in tracing the early papers that have served as the foundations of the field. Cheret [Shock Compression of Condensed Matter - 1989, Elsevier Sci. Pub. B. V. , pp 11-19, 1990) has contributed to this literature with his review of the life of Hugoniot on the one-hundredth anniversary of the publication of Hugoniot's classic paper on the propagation of discontinuous waves in gases. This contribution prompted additional historical investigation involving the precursors to Hugoniot.

Der Band enthalt den Abschlussbericht des DFG-Schwerpunktprogramms "Flusssimulation mit Hochstleistungsrechnern." Es fuhrt die Arbeiten fort, die schon als Band 38 in der Reihe "Notes on Numerical Fluid Mechanics" erschienen sind.Work is reported, which was sponsored by the Deutsche Forschungsgemeinschaft from 1993 to 1995. Scientists from numerical mathematics, fluid mechanics, aerodynamics, and turbomachinery present their work on flow simulation with massively parallel systems, on the direct and large-eddy simulation of turbulence, and on mathematical foundations, general solution techniques and applications. Results are reported from benchmark computations of laminar flow around a cylinder, in which seventeen groups participated."

Large-Eddy Simulation (LES), which is an advanced eddy-resolving method for calculating turbulent flows, is used increasingly in Computational Fluid Dynamics, also for solving hydraulics and environmental flow problems. The method has generally great potential and is particularly suited for problems dominated by large-scale turbulent structures. This book gives an introduction to the LES method specially geared for hydraulic and environmental engineers. Compared with existing books on LES it is less theoretically and mathematically demanding and hence easier to follow, and it covers special features of flows in water bodies and summarizes the experience gained with LES for calculating such flows. The book was written primarily as an introduction to LES for hydraulic and environmental engineers, but it will also be very useful as an entry to the subject of LES for researchers and students in all fi elds of fl uids engineering. The applications part will further be useful to researchers interested in the physics of fl ows governed by the dynamics of coherent structures.

The primary purpose of this text is to document many of the lessons that have been learned during the author s more than forty years in the field of blast and shock. The writing therefore takes on an historical perspective, in some sense, because it follows the author s experience. The book deals with blast waves propagating in fluids or materials that can be treated as fluids.

It begins by distinguishing between blast waves and the more general category of shock waves. It then examines several ways of generating blast waves, considering the propagation of blast waves in one, two and three dimensions as well as through the real atmosphere. One section treats the propagation of shocks in layered gases in a more detailed manner.

The book also details the interaction of shock waves with structures in particular reflections, progressing from simple to complex geometries, including planar structures, two-dimensional structures such as ramps or wedges, reflections from heights of burst, and three-dimensional structures.

Intended for those with a basic knowledge of algebra and a solid grasp of the concepts of conservation of mass and energy, the text includes an introduction to blast wave terminology and conservation laws as well as a discussion of units and the importance of consistency."

Some words about SCART 2000. SCART stands for science and art. SCART meetings are organized in a loose time sequence by an international group of scientists, most of them fluid-dynamicists. The first meeting was held in Hong-Kong, the second one in Berlin, and the third, and latest, one in Zurich. SCART meetings include a scientific conference and a number of art events. The intention is to restart a dialogue between scientists and artists which was so productive in the past. To achieve this goal several lectures given by scientists at the conference are intended for a broader public. In the proceedings they are denoted as SCART lectures. The artists in tum address the main theme of the conference with their contributions. The lectures at SCART 2000 covered the entire field of fluiddynamics, from laminar flows in biological systems to astrophysical events, such as the explosion of a neutron star. The main exhibition by Dutch and Swiss artists showed video and related art under the title 'Walking on Air'. Experimental music was performed in two concerts.

Aircraft concepts are always driven by the requirements of the desired m- sion. A di?erent purpose for the use of the aircraft consequently results in a di?erent design. Therefore, depending on the intended outcome, con?i- ing requirements need to be ful?lled, for example, e?cient cruise speed and greatercargocapabilities,in combinationwith shorttake-o?andlanding ?eld lengths, or high speed and agility combined with variable payload demands. Due to the highly complex, non-linear physical environment in which aircraft operate, this task demands that the most advanced methods and tools are employed, to gain the necessary understanding of ?ow phenomena, and to exploit the ?ow physics to achieve maximum aircraft e?ciency. Inthe naturalsciences,researcherstry to create andextend humankno- edge by understanding and explaining the mechanisms of physical processes. In engineering, a designer is limited by certain requirements, and in order to ful?l these requirements the necessary technical tools need to be designed. In general, for a given problem the corresponding scienti?c or technical solution is sought. In order to successfully advance from a problem towards a solution, three main methods may be used. The two classical methods include theory and experiment, which are now being complemented by a third method, - scribedasnumericalsimulation.Theexperimentalapproachis basedonph- ical observation, measurement of relevant values, and methodical variation of the subject matter. For example, such experiments are used to gain a ph- ical understanding as well as to validate and investigate design alternatives.

The homogenization of single phase gases or liquids with chemical reactive components by mixing belongs to one of the oldest basic operations applied in chemical engineering. The design of equipment for mixing processes is still derived from measurements of the mixing time which is related to the applied methods of measurement and the special design of the test equipment itself.

This book was stimulated by improved modern methods for experimental research and visualization, for simulations and numerical calculations of mixing and chemical reactions in micro and macro scale of time and local coordinates. It is aimed to improve the prediction of efficiencies and selectivities of chemical reactions in macroscopic scale. The results should give an understanding of the influence of the construction of different mixing equipment on to the momentum, heat and mass transfer as well as reaction processes running on microscopic scales of time and local coordinates.

Newly developed methods of measurement are adjusted to the scales of the selected special transport and conversion processes. They allow a more detailed modeling of the mixing processes by the formulation of an appropriate set of momentum-, heat- and mass balance equations as well as boundary conditions in time and local coordinates together with constitutive equations and reaction kinetics equations as closure laws for numerical and analytical calculations. The latter were empirically derived in the past and therefore of limited reliability only.

The improved and more detailed modeling leads to a major progress in predicting mixing processes on the different scales adjusted to transport and reaction processes in molecular, micro- and macro dimensions.

As a consequence improved numerical calculations are performed on the basis of newly derived experimental, measurement and modeling methods which are the basis for the prediction of mixing time as well as conversion rates and selectivities of chemical reactions during the mixing process. The research efforts are focused onto the design of the technical equipment for flow mixing processes. Mixing is performed inside velocity fields leading to deformation gradients from free or wall induced boundary layers. The different kinds of process equipment are jet mixer, static mixer and mixing vessels equipped with rotating stirrers. Especially in micro mixing newly developed constructions are investigated permitting the scale up from laboratory to technical dimensions.

The origins of turbulent ?ow and the transition from laminar to turbulent ?ow are the most important unsolved problems of ?uid mechanics and aerodynamics. - sides being a fundamental question of ?uid mechanics, there are numerous app- cations relying on information regarding transition location and the details of the subsequent turbulent ?ow. For example, the control of transition to turbulence is - pecially important in (1) skin-friction reduction of energy ef?cient aircraft, (2) the performance of heat exchangers and diffusers, (3) propulsion requirements for - personic aircraft, and (4) separation control. While considerable progress has been made in the science of laminar to turbulent transition over the last 30 years, the c- tinuing increase in computer power as well as new theoretical developments are now revolutionizing the area. It is now starting to be possible to move from simple 1D eigenvalue problems in canonical ?ows to global modes in complex ?ows, all - companied by accurate large-scale direct numerical simulations (DNS). Here, novel experimental techniques such as modern particle image velocimetry (PIV) also have an important role. Theoretically the in?uence of non-normality on the stability and transition is gaining importance, in particular for complex ?ows. At the same time the enigma of transition in the oldest ?ow investigated, Reynolds pipe ?ow tran- tion experiment, is regaining attention. Ideas from dynamical systems together with DNS and experiments are here giving us new insights.

This book contains contributions by colleagues, former students and friends of Professor Eli Reshotko in celebration of his 60th birth day. Since Professor Reshotko's scientific and engineering contribu tions have been in the areas of hydrodynamic stability, transition to turbulence, and boundary layer flows, it is only appropriate that the articles in this volume be devoted to these and related topics. The first two sections focus on instabilities and transition in sub sonic and supersonic flows, respectively. The third section deals with developing turbulence, while the the final section treats related prob lems in engineering fluid mechanics. The diversity and scope of the articles contained herein exemplify the insight and expertise required in the study of transitional and turbulent flows today - traits which also exemplify Eli Reshotko's contributions to these fields. A few of the articles in this volume were presented at a sym posium in honor of Eli Reshotko's 60th birthday, held in Newport News, Virginia, on July 28, 1991. The symposium was sponsored by lCASE, and organized by M.Y. Hussaini (lCASE) and R. Hirsh (U.S. National Science Foundation). Of those who could not attend, many chose to honor Professor Reshotko by a contribution to the volume dedicated to him. We would like to use this opportunity to express our deep ap preciation to M.Y. Hussaini for initiating this very special tribute to Eli, and to Ms. Emily Todd for her efforts in the volume preparation and in the organization of the symposium."

This title provides the fundamental bases for developing turbulence models on rational grounds. The main different methods of approach are considered, ranging from statistical modelling at various degrees of complexity to numerical simulations of turbulence. Each of these various methods has its own specific performances and limitations, which appear to be complementary rather than competitive. After a discussion of the basic concepts, mathematical tools and methods for closure, the book considers second order closure models.Emphasis is placed upon this approach because it embodies potentials for clarifying numerous problems in turbulent shear flows. Simpler, generally older models are then presented as simplified versions of the more general second order models. The influence of extra physical parameters is also considered. Finally, the book concludes by examining large Eddy numerical simulations methods. Given the book's comprehensive coverage, those involved in the theoretical or practical study of turbulence problems in fluids will find this a useful and informative read.

Over the last decade, Computational Fluid Dynamics (CFD) has become a - ture technology for the development of new products in aeronautical industry. Aerodynamic design engineers have progressively taken advantage of the pos- bilities o?ered by the numericalsolutionof the Reynolds averagedNavier-Stokes (RANS) equations. Signi?cant improvements in physical modeling and solution algorithms as well as the enormous increase of computer power enable hi- ?delity numerical simulations in all stages of aircraft development. In Germany, the national CFD project MEGAFLOW furthered the dev- opment and availability of RANS solvers for the prediction of complex ?ow problemssigni?cantly. MEGAFLOWwasinitiated by the?rstaviationresearch programoftheFederalGovernmentin1995undertheleadershipoftheDLR(see Kroll, N. , Fassbender, J. K. (Eds). : MEGAFLOW - Numerical Flow Simulation for Aircraft Design; Notes on Numerical Fluid Mechanics and Multidisciplinary Design, Volume 89, Springer, 2005). A network from aircraft industry, DLR and several universities was created with the goal to focus and direct development activities for numerical ?ow simulation towards a common aerodynamic si- lation system providing both a block-structured (FLOWer-Code) and a hybrid (TAU-Code) parallel ? ow prediction capability. Today, both codes have reached a high level of maturity and reliability. They are routinely used at DLR and German aeronautic industry for a wide range of aerodynamic applications. For many universities the MEGAFLOW software represents a platform for the - provementofphysicalmodelsandfortheinvestigationofcomplex?owproblems. The network was established as an e?cient group of very closely co-operating partners with supplementing expertises and experience.

This short but complicated book is very demanding of any reader. The scope and style employed preserve the nature of its subject: the turbulence phe nomena in gas and liquid flows which are believed to occur at sufficiently high Reynolds numbers. Since at first glance the field of interest is chaotic, time-dependent and three-dimensional, spread over a wide range of scales, sta tistical treatment is convenient rather than a description of fine details which are not of importance in the first place. When coupled to the basic conserva tion laws of fluid flow, such treatment, however, leads to an unclosed system of equations: a consequence termed, in the scientific community, the closure problem. This is the central and still unresolved issue of turbulence which emphasizes its chief peculiarity: our inability to do reliable predictions even on the global flow behavior. The book attempts to cope with this difficult task by introducing promising mathematical tools which permit an insight into the basic mechanisms involved. The prime objective is to shed enough light, but not necessarily the entire truth, on the turbulence closure problem. For many applications it is sufficient to know the direction in which to go and what to do in order to arrive at a fast and practical solution at minimum cost. The book is not written for easy and attractive reading."