This is Volume 4 of the book series of the Body and Soul mathematics education reform program. It presents a unified new approach to computational simulation of turbulent flow starting from the general basis of calculus and linear algebra of Vol 1-3. The book puts the Body and Soul computational finite element methodology in the form of General Galerkin (G2) up against the challenge of computing turbulent solutions of the inviscid Euler equations and the Navier-Stokes equations with small viscosity. This is an outstanding textbook presenting plenty of new material with an excellent pedagogical approach.

Current research fields in science and technology were presented and discussed at the EKC2008, informing about the interests and directions of the scientists and engineers in EU countries and Korea. The Conference has emerged from the idea of bringing together EU and Korea to get to know each other better, especially in fields of science and technology. The focus of the conference is put on the topics: Computational Fluid Dynamics; Mechatronics and Mechanical Engineering; Information and Communications Technology; Life and Natural Sciences; Energy and Environmental Technology.

Time-Resolved Spectroscopy in Complex Liquids is intended to introduce the experimental researchers to state-of-the-art techniques in the study of the dynamics of complex liquids. The contributors concentrate on time-resolved optical spectroscopy, which recently produced many relevant results and new information about complex liquids. This is an emerging topic of soft-matter science and this book provides the most up-to-date account of new development.

Homogenization is not about periodicity, or Gamma-convergence, but about understanding which effective equations to use at macroscopic level, knowing which partial differential equations govern mesoscopic levels, without using probabilities (which destroy physical reality); instead, one uses various topologies of weak type, the G-convergence of Sergio Spagnolo, the H-convergence of Francois Murat and the author, and some responsible for the appearance of nonlocal effects, which many theories in continuum mechanics or physics guessed wrongly.

For a better understanding of 20th century science, new mathematical tools must be introduced, like the author's H-measures, variants by Patrick Gerard, and others yet to be discovered. "

This book was written as a graduate student course--Shock Dynamics. Up to now, the first author has taught this course to the graduate students in the field of Fluid Mechanics, Department of Modern Mechanics, University of Science and Technology of China for seven times. In the spring semester 1989, during his visit to the United States, the first author taught this course to the graduate students of Department of Mathemat ics, University of Colorado at Denver. At the same time, he gave a series of four lectures on Shock Dynamics to the graduate students of Department of Aerospace Engineering Sciences, University of Colorado at Boulder. In 1991, during the first author's visit to Japan, he gave some lectures on Shock Dynamics in Tohoku University, University of Tokyo and Kyushu Uni versity. The dynamic phenomena of shock waves such as propagation, diffraction, reflection, refraction and interaction of shock waves may be studied by using experimental methods, numerical calculations and theoretical analyses. Although the detailed flow patterns of phenomena of shock motion can be obtained by using experimental methods and numerical calculations of solving Euler Equation or Navier-Stokes Equation, for example, the diffractions of shock waves by wedges form various phenomena of reflection--RR, SMR, CMR and DMR, we also need to analyse the process of the formation of shock waves in various phenomena of diffraction, reflection and interaction by using theoretical methods."

Preface "In aircraft design, efficiency is determined by the ability to accurately and rel- bly predict the occurrence of, and to model the development of, turbulent flows. Hence, the main objective in industrial computational fluid dynamics (CFD) is to increase the capabilities for an improved predictive accuracy for both complex flows and complex geometries." This text part taken from Haase et al (2006), - scribing the results of the DESider predecessor project "FLOMANIA" is still - and will be in future valid. With an ever-increasing demand for faster, more reliable and cleaner aircraft, flight envelopes are necessarily shifted into areas of the flow regimes exhibiting highly unsteady and, for military aircraft, unstable flow behaviour. This undou- edly poses major new challenges in CFD; generally stated as an increased pred- tive accuracy whist retaining "affordable" computation times. Together with highly resolved meshes employing millions of nodes, numerical methods must have the inherent capability to predict unsteady flows. Although at present, (U)RANS methods are likely to remain as the workhorses in industry, the DESider project focussed on the development and combination of these approaches with LES methods in order to "bridge" the gap between the much more expensive (due to high Reynolds numbers in flight), but more accurate (full) LES.

An exciting new direction in hydrodynamic stability theory and the transition to turbulence is concerned with the role of disconnected states or finite amplitude solutions in the evolution of disorder in fluid flows. This volume contains refereed papers presented at the IUTAM/LMS sponsored symposium on "Non-Uniqueness of Solutions to the Navier-Stokes equations and their Connection with Laminar-Turbulent Transition" held in Bristol 2004. Theoreticians and experimentalists gathered to discuss developments in understanding both the onset and collapse of disordered motion in shear flows such as those found in pipes and channels.

The central objective of the symposium was to discuss the increasing amount of experimental and numerical evidence for finite amplitude solutions to the Navier-Stokes equations and to set the work into a modern theoretical context. The participants included many of the leading authorities in the subject and this volume captures much of the flavour of the resulting stimulating and lively discussions.

In order to allow the application of the theory from all the three volumes also to processes in combustion engines a systematic set of internally consistent state equations for diesel fuel gas and liquid valid in broad range of changing pressure and temperature are provided also in Volume 3. Erlangen, October 2006 Nikolay Ivanov Kolev Table of contents 1 Some basics of the single-phase boundary layer theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. 1 Flow over plates, velocity profiles, share forces, heat transfer. . . . . . . . . . . . . . . . . . . . 1 1. 1. 1 Laminar flow over the one site of a plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. 1. 2 Turbulent flow parallel to plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. 2 Steady state flow in pipes with circular cross sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1. 2. 1 Hydraulic smooth wall surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1. 2. 2 Transition region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1. 2. 3 Complete rough region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1. 2. 4 Heat transfer to fluid in a pipe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1. 3 Transient flow in pipes with circular cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Nomenclature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 Introduction to turbulence of multi-phase flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2. 1 Basic ideas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2. 2 Isotropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2. 3 Scales, eddy viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2. 3. 1 Small scale turbulent motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2. 3. 2 Large scale turbulent motion, Kolmogorov-Pandtl expression. . . . . . . . . 42 2. 4 k-eps framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Nomenclature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3 Sources for fine resolution outside the boundary layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3. 1 Bulk sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3. 1. 1 Deformation of the velocity field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3. 1. 2 Blowing and suction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The importance of vorticity and vortex dynamics has now been well rec- nized at both fundamental and applied levels of ?uid dynamics, as already anticipatedbyTruesdellhalfcenturyagowhenhewrotethe?rstmonograph onthesubject, The Kinematics of Vorticity(1954);andasalsoevidencedby the appearance of several books on this ?eld in 1990s. The present book is characterizedbythefollowingfeatures: 1. A basic physical guide throughout the book. The material is directed by a basic observation on the splitting and coupling of two fundamental processes in ?uid motion, i.e., shearing (unique to ?uid) and compre- ing/expanding.Thevorticityplaysakeyroleintheformer, andavortex isnothingbuta?uidbodywithhighconcentrationofvorticitycompared to its surrounding ?uid. Thus, the vorticity and vortex dynamics is - cordinglyde?nedasthetheoryofshearingprocessanditscouplingwith compressing/expandingprocess. 2. A description of the vortex evolution following its entire life.Thisbegins from the generation of vorticity to the formation of thin vortex layers andtheirrolling-upintovortices, fromthevortex-corestructure, vortex motionandinteraction, totheburstofvortexlayerandvortexintosma- scalecoherentstructureswhichleadstothetransitiontoturbulence, and ?nallytothedissipationofthesmalleststructuresintoheat. 3. Wide range of topics.Inadditiontofundamentaltheoriesrelevanttothe abovesubjects, theirmostimportantapplicationsarealsopresented.This includes vortical structures in transitional and turbulent ?ows, vortical aerodynamics, and vorticity and vortices in geophysical ?ows. The last topic was suggested to be added by Late Sir James Lighthill, who read carefullyanearlydraftoftheplannedtableofcontentsofthebookin1994 andexpressedthathelikes"allthematerial"thatweproposedthere. These basic features of the present book are a continuation and - velopment of the spirit and logical structure of a Chinese monograph by the same authors, Introduction to Vorticity and Vortex Dynamics, Higher VI Preface EducationPress, Beijing,1993, butthematerialhasbeencompletelyrewr- tenandupdated.Thebookmay?tvariousneedsof?uiddynamicsscientists, educators, engineers, aswellasappliedmathematicians.Itsselectedchapters canalsobeusedastextbookforgraduatestudentsandseniorundergraduates. Thereadershouldhaveknowledgeofundergraduate?uidmechanicsand/or aerodynamicscourses.

This volume presents a review of advanced technological problems in the glass industry and of the mathematics involved. It is amazing that such a seemingly small research area is extremely rich and calls for an impressively large variety of mathematical methods, including numerical simulations of considerable complexity. The problems treated here are very typical of the field of glass manufacturing and cover a large spectrum of complementary subjects: injection molding by various techniques, radiative heat transfer in glass, nonisothermal flows and fibre spinning. The book can certainly be useful not only to applied mathematicians, but also to physicists and engineers, who can find in it an overview of the most advanced models and methods.

Wave Turbulence refers to the statistical theory of weakly nonlinear dispersive waves. There is a wide and growing spectrum of physical applications, ranging from sea waves, to plasma waves, to superfluid turbulence, to nonlinear optics and Bose-Einstein condensates. Beyond the fundamentals the book thus also covers new developments such as the interaction of random waves with coherent structures (vortices, solitons, wave breaks), inverse cascades leading to condensation and the transitions between weak and strong turbulence, turbulence intermittency as well as finite system size effects, such as "frozen" turbulence, discrete wave resonances and avalanche-type energy cascades. This book is an outgrow of several lectures courses held by the author and, as a result, written and structured rather as a graduate text than a monograph, with many exercises and solutions offered along the way. The present compact description primarily addresses students and non-specialist researchers wishing to enter and work in this field.

Computational Fluid Dynamics: An Introduction grew out of a von Karman Institute (VKI) Lecture Series by the same title ?rst presented in 1985 and repeated with modi?cations every year since that time. The objective, then and now, was to present the subject of computational ?uid dynamics (CFD) to an audience unfamiliar with all but the most basic numerical techniques and to do so in such a way that the practical application of CFD would become clear to everyone. A second edition appeared in 1995 with updates to all the chapters and when that printing came to an end, the publisher requested that the editor and authors consider the preparation of a third edition. Happily, the authors received the request with enthusiasm. The third edition has the goal of presenting additional updates and clari?cations while preserving the introductory nature of the material. The book is divided into three parts. John Anderson lays out the subject in Part I by ?rst describing the governing equations of ?uid dynamics, concentrating on their mathematical properties which contain the keys to the choice of the numerical approach. Methods of discretizing the equations are discussed and transformation techniques and grids are presented. Two examples of numerical methods close out this part of the book: source and vortex panel methods and the explicit method. Part II is devoted to four self-contained chapters on more advanced material. Roger Grundmann treats the boundary layer equations and methods of solution.

The 24 papers presented at the international concluding colloquium of the German priority programme (DFG-Verbundschwerpunktprogramm) "Transition," held in April 2002 in Stuttgart.

The unique and successful programme ran six years, starting April 1996, and was sponsored mainly by the Deutsche Forschungsgemeinschaft, DFG, but also by the Deutsches Zentrum f r Luft-und Raumfahrt, DLR, the Physikalisch-Technische Bundesanstalt Braunschweig, PTB, and Airbus Deutschland. The papers summarise the results of the programme and cover transition mechanisms, transition prediction, transition control, natural transition and measurement techniques, transition - turbulence - separation, and visualisation issues. Three invited papers are devoted to mechanisms of turbulence production, to a general framework of stability, receptivity and control, and a forcing model for receptivity analysis. Almost every transition topic arising in subsonic and transonic flow is covered.

Although the solution of Partial Differential Equations by numerical methods is the standard practice in industries, analytical methods are still important for the critical assessment of results derived from advanced computer simulations and the improvement of the underlying numerical techniques. Literature devoted to analytical methods, however, often focuses on theoretical and mathematical aspects and is therefore useless to most engineers. Analytical Methods for Heat Transfer and Fluid Flow Problems addresses engineers and engineering students. It describes useful analytical methods by applying them to real-world problems rather than solving the usual over-simplified classroom problems. The book demonstrates the applicability of analytical methods even for complex problems and guides the reader to a more intuitive understanding of approaches and solutions.

The sixth ERCOFTAC Workshop on Direct and Large-Eddy Simulation (DLES-6) was held at the University of Poitiers from September 12-14, 2005. Following the tradition of previous workshops in the DLES-series, this edition has reflected the state-of-the-art of numerical simulation of transitional and turbulent flows and provided an active forum for discussion of recent developments in simulation techniques and understanding of flow physics."

This volume contains the proceedings of the 2001 DLES4 workshop. It describes and discusses state-of-the-art modeling and simulation approaches for complex flows. Fundamental turbulence and modeling issues but also elements from modern numerical analysis are at the heart of this field of interest.

This book contains 23 papers presented at the ECCOMAS Multidisciplinary Jubilee Symposium - New Computational Challenges in Materials, Structures, and Fluids (EMJS08), in Vienna, February 18-20, 2008. The main intention of EMJS08 was to react adequately to the increasing need for interdisciplinary research activities allowing ef?cient solution of complex problems in engineering and in the applied sciences. The 15th anniversary of ECCOMAS (European Community on Computational Methods in Applied Sciences) provided a suitable frame for taking the afo- mentioned situation into account by inviting distinguished colleagues from d- ferent areas of engineering and the applied sciences, encouraging them to choose multidisciplinary topics for their lectures. The main themes of EMJS08 have a long tradition in engineering and in the applied sciences: materials, structures, and ?uids. The solution of scienti?c pr- lems involving ?uids together with solids and structures, not to forget the materials the structures are made of, is of paramount importance in a technical world of rapidly increasing sophistication, referred to as the Leonardo World by the eminent German philosopher Jurgen Mittelstrass. More recently, the main themes of EMJS08 have gained considerable mom- tum, owing to signi?cant progress in nanotechnology. It enables resolution of a multitude of materials into their micro- and nanostructures. Covering aspects such as * Physical and chemical characterization * Multiscale modeling concepts, continuum micromechanics, and computational homogenization, as well as * Applications in various engineering ?elds the individual contributions to this book ?ow along different tracks of ?uids, materials, and structures.

Structures in contact with fluid flow, whether natural or man-made, are inevitably subject to flow-induced forces and flow-induced vibration: from plant leaves to traffic signs and to more substantial structures, such as bridge decks and heat exchanger tubes. Under certain conditions the vibration may be self-excited, and it is usually referred to as an instability. These instabilities and, more specifically, the conditions under which they arise are of great importance to designers and operators of the systems concerned because of the significant potential to cause damage in the short term. Such flow-induced instabilities are the subject of this book. In particular, the flow-induced instabilities treated in this book are associated with cross-flow, that is, flow normal to the long axis of the structure. The book treats a specific set of problems that are fundamentally and technologically important: galloping, vortex-shedding oscillations under lock-in conditions and rain-and-wind-induced vibrations, among others.

Everything important, up-to-date and practical about turbopumps can be found in this book. The material is arranged to cover the most important topics, from basic theories to practical applications. This book can also serve as a useful textbook for students who are taking courses in the area of turbopumps and hydraulic machineries. It is the complete reference book for turbopumps.

Microfluidics for Biological Applications provides researchers and scientists in the biotechnology, pharmaceutical, and life science industries with an introduction to the basics of microfluidics and also discusses how to link these technologies to various biological applications at the industrial and academic level. Readers will gain insight into a wide variety of biological applications for microfluidics.

The material presented here is divided into four parts, Part I gives perspective on the history and development of microfluidic technologies, Part II presents overviews on how microfluidic systems have been used to study and manipulate specific classes of components, Part III focuses on specific biological applications of microfluidics: biodefense, diagnostics, high throughput screening, and tissue engineering and finally Part IV concludes with a discussion of emerging trends in the microfluidics field and the current challenges to the growth and continuing success of the field.

In this authoritative and comprehensive volume, Claude Bardos and Andrei Fursikov have drawn together an impressive array of international contributors to present important recent results and perspectives in this area. The main subjects that appear here relate largely to mathematical aspects of the theory but some novel schemes used in applied mathematics are also presented. Various topics from control theory, including Navier-Stokes equations, are covered.

This book covers specific aspects of submarine hydrodynamics in a very practical manner. The author reviews basic concepts of ship hydrodynamics and goes on to show how they are applied to submarines, including a look at the use of physical model experiments. The book is intended for professionals working in submarine hydrodynamics, as well as for advanced students in the field. This revised edition includes updated information on empirical methods for predicting the hydrodynamic manoeuvring coefficients, and for predicting the resistance of a submarine. It also includes new material on how to assess propulsors, and includes measures of wake distortion, which has a detrimental influence on propulsor performance. Additional information on safe manoeuvring envelopes is also provided. The wide range of references has been updated to include the latest material in the field.

A few years ago the Helmholtz Association (HGF) consisting of 15 research Institutions including the German Aerospace Center (DLR) started a network research program called 'Virtual Institutes'. The basic idea of this program was to establish research groups formed by Helmholtz research centers and universities to study and develop methods or technologies for future applications and educate young scientists. It should also enable and encourage the partners of this Virtual Institute after 3 years funding to continue their cooperation in other programs. Following this HGF request and chance the DLR Windtunnel Department of the Institute of Aerodynamics and Flow Technology took the initiative and established a network with other DLR institutes and German u- versities RWTH Aachen, University of Stuttgart and Technical University Munich. The main goal of this network was to share the experience in system analysis, ae- dynamics and material science for aerospace for improving the understanding and applicability of some key technologies for future reusable space transportation s- tems. Therefore, the virtual institute was named RESPACE (Key Technologies for Re- Usable Space Systems).

The aerospace industry increasingly relies on advanced numerical simulation tools in the early design phase. This volume provides the results of a German initiative which combines many of the CFD development activities from the German Aerospace Center (DLR), universities, and aircraft industry. Numerical algorithms for structured and hybrid Navier-Stokes solvers are presented in detail. The capabilities of the software for complex industrial applications are demonstrated.

In a typical mathematical model of a controlled distributed parameter process one usually ?nds either boundary or internal locally distributed controls to serve as the means to describe the effect of external actuators on the process at hand. H- ever, these classical controls, enteringthe modelequationsas additive terms, are not suitable to deal with a vast array of processes that can change their principal intr- sic properties due to the control actions. Important examples here include (but not limitedto)thechainreaction-typeprocessesinbiomedical, nuclear, chemicalan- nancial applications, which can changetheir (reaction)rate when certain "catalysts" are applied, and the so-called "smart materials," which can, for instance, alter their frequency response. The goal of this monograph is to address the issue of global controllability of partial differential equations in the context of multiplicative (or bilinear) c- trols, which enter the model equations as coef?cients. The mathematical models of our interest include the linear and nonlinear parabolic and hyperbolic PDE's, the Schrodi ] nger equation, and coupled hybrid nonlinear distributed parameter systems associated with the swimming phenomenon. Pullman, WA, USA Alexander Khapalov January 2010 vii Preface This monograph developed from the research conducted in 2001-2009 in the area of controllability theory of partial differential equations. The concept of control- bility is a principal component of Control Theory which was brought to life in the 1950's by numerous applications in engineering, and has received the most sign- icant attention both from the engineering and the mathematical communities since then."