This book is dedicated to the mathematical study of two-dimensional statistical hydrodynamics and turbulence, described by the 2D Navier-Stokes system with a random force. The authors' main goal is to justify the statistical properties of a fluid's velocity field u(t,x) that physicists assume in their work. They rigorously prove that u(t,x) converges, as time grows, to a statistical equilibrium, independent of initial data. They use this to study ergodic properties of u(t,x) - proving, in particular, that observables f(u(t,.)) satisfy the strong law of large numbers and central limit theorem. They also discuss the inviscid limit when viscosity goes to zero, normalising the force so that the energy of solutions stays constant, while their Reynolds numbers grow to infinity. They show that then the statistical equilibria converge to invariant measures of the 2D Euler equation and study these measures. The methods apply to other nonlinear PDEs perturbed by random forces.

This volume reports results from the German research initiative MUNA (Management and Minimization of Errors and Uncertainties in Numerical Aerodynamics), which combined development activities of the German Aerospace Center (DLR), German universities and German aircraft industry. The main objective of this five year project was the development of methods and procedures aiming at reducing various types of uncertainties that are typical of numerical flow simulations. The activities were focused on methods for grid manipulation, techniques for increasing the simulation accuracy, sensors for turbulence modelling, methods for handling uncertainties of the geometry and grid deformation as well as stochastic methods for quantifying aleatoric uncertainties.

In this monograph we present a review of a number of recent results on the motion of a classical body immersed in an infinitely extended medium and subjected to the action of an external force. We investigate this topic in the framework of mathematical physics by focusing mainly on the class of purely Hamiltonian systems, for which very few results are available. We discuss two cases: when the medium is a gas and when it is a fluid. In the first case, the aim is to obtain microscopic models of viscous friction. In the second, we seek to underline some non-trivial features of the motion. Far from giving a general survey on the subject, which is very rich and complex from both a phenomenological and theoretical point of view, we focus on some fairly simple models that can be studied rigorously, thus providing a first step towards a mathematical description of viscous friction. In some cases, we restrict ourselves to studying the problem at a heuristic level, or we present the main ideas, discussing only some aspects of the proof if it is prohibitively technical. This book is principally addressed to researchers or PhD students who are interested in this or related fields of mathematical physics.

The purpose of this book is to present the basic elements of numerical methods for compressible flows. It is appropriate for advanced undergraduate and graduate students and specialists working in high speed flows. The focus is on the unsteady one-dimensional Euler equations which form the basis for numerical algorithms in compressible fluid mechanics. The book is restricted to the basic concepts of finite volume methods, and even in this regard is not intended to be exhaustive in its treatment. Although the practical applications of the one-dimensional Euler equations are limited, virtually all numerical algorithms for inviscid compressible flow in two and three dimensions owe their origin to techniques developed in the context of the one-dimensional Euler equations. The author believes it is therefore essential to understand the development and implementation of these algorithms in their original one-dimensional context. The text is supplemented by numerous end-of-chapter exercises.

Modelling and Simulation of Reactive Flows presents information on modeling and how to numerically solve reactive flows. The book offers a distinctive approach that combines diffusion flames and geochemical flow problems, providing users with a comprehensive resource that bridges the gap for scientists, engineers, and the industry. Specifically, the book looks at the basic concepts related to reaction rates, chemical kinetics, and the development of reduced kinetic mechanisms. It considers the most common methods used in practical situations, along with equations for reactive flows, and various techniques-including flamelet, ILDM, and Redim-for jet flames and plumes, with solutions for both. In addition, the book includes techniques to accelerate the convergence of numerical simulation, and a discussion on the analysis of uncertainties with numerical results, making this a useful reference for anyone who is interested in both combustion in free flow and in porous media.

This book presents recent results on nonlinear evolutionary fluid equations such as the compressible (radiative) magnetohydrodynamics (MHD) equations, compressible viscous micropolar fluid equations, the full non-Newtonian fluid equations and non-autonomous compressible Navier-Stokes equations. It summarizes recently published research by the authors and their colleagues and also includes new and unpublished material. This type of partial differential equations arises in many fields of mathematics, but also in other branches of science such as physics and fluid dynamics. This book will be a valuable resource for graduate students and researchers interested in partial differential equations, and will also benefit practitioners in physics and engineering.

This book introduces key ideas and principles in the theory of elasticity with the help of symbolic computation. Differential and integral operators on vector and tensor fields of displacements, strains and stresses are considered on a consistent and rigorous basis with respect to curvilinear orthogonal coordinate systems. As a consequence, vector and tensor objects can be manipulated readily, and fundamental concepts can be illustrated and problems solved with ease. The method is illustrated using a variety of plane and three-dimensional elastic problems. General theorems, fundamental solutions, displacements and stress potentials are presented and discussed. The Rayleigh-Ritz method for obtaining approximate solutions is introduced for elastostatic and spectral analysis problems. Containing more than 60 exercises and solutions in the form of Mathematica notebooks that accompany every chapter, the reader can learn and master the techniques while applying them to a large range of practical and fundamental problems.

This book was first published in 2005. When an oceanic tidal wave that is primarily active on the water surface passes an ocean shelf or a region with a seamount, it is split into a less energetic surface wave and other internal modes with different wavelengths and propagation speeds. This cascading process, from the barotropic tides to the baroclinic components, leads to the transformation of tidal energy into turbulence and heat, an important process for the dynamics of the lower ocean. Baroclinic Tides demonstrates the analytical and numerical methods used to study the generation and evolution of baroclinic tides and, by comparison with experiments and observational data, shows how to distinguish and interpret internal waves. Strongly non-linear solitary internal waves, which are generated by internal tidal waves at the final stage of their evolution, are investigated in detail. This book is intended for researchers and graduate students of physical oceanography, geophysical fluid dynamics and hydroacoustics.

The problems of making inferences about the natural world from noisy observations and imperfect theories occur in almost all scientific disciplines. This 2006 book addresses these problems using examples taken from geophysical fluid dynamics. It focuses on discrete formulations, both static and time-varying, known variously as inverse, state estimation or data assimilation problems. Starting with fundamental algebraic and statistical ideas, the book guides the reader through a range of inference tools including the singular value decomposition, Gauss-Markov and minimum variance estimates, Kalman filters and related smoothers, and adjoint (Lagrange multiplier) methods. The final chapters discuss a variety of practical applications to geophysical flow problems. Discrete Inverse and State Estimation Problems is an ideal introduction to the topic for graduate students and researchers in oceanography, meteorology, climate dynamics, and geophysical fluid dynamics. It is also accessible to a wider scientific audience; the only prerequisite is an understanding of linear algebra.

This volume contains the contributions to the 17th Symposium of STAB (German Aerospace Aerodynamics Association). STAB includes German scientists and engineers from universities, research establishments and industry doing research and project work in numerical and experimental fluid mechanics and aerodynamics, mainly for aerospace but also for other applications. Many of the contributions collected in this book present results from national and European Community sponsored projects. This volume gives a broad overview of the ongoing work in this field in Germany and spans a wide range of topics: airplane aerodynamics, multidisciplinary optimization and new configurations, hypersonic flows and aerothermodynamics, flow control (drag reduction and laminar flow control), rotorcraft aerodynamics, aeroelasticity and structural dynamics, numerical simulation, experimental simulation and test techniques, aeroacoustics as well as the new fields of biomedical flows, convective flows, aerodynamics and acoustics of high-speed trains.

Multi-phase flows are part of our natural environment such as tornadoes, typhoons, air and water pollution and volcanic activities as well as part of industrial technology such as power plants, combustion engines, propulsion systems, or chemical and biological industry. The industrial use of multi-phase systems requires analytical and numerical strategies for predicting their behavior. .In its fourth extended edition the successful monograph package "Multiphase Flow Daynmics" contains theory, methods and practical experience for describing complex transient multi-phase processes in arbitrary geometrical configurations, providing a systematic presentation of the theory and practice of numerical multi-phase fluid dynamics. In the present second volume the methods for describing the mechanical interactions in multiphase dynamics are provided. This fourth edition includes various updates, extensions, improvements and corrections. "The literature in the field of multiphase flows is numerous. Therefore, it is very important to have a comprehensive and systematic overview including useful numerical methods. The volumes have the character of a handbook and accomplish this function excellently. The models are described in detail and a great number of comprehensive examples and some cases useful for testing numerical solutions are included. These two volumes are very useful for scientists and practicing engineers in the fields of technical thermodynamics, chemical engineering, fluid mechanics, and for mathematicians with interest in technical problems. Besides, they can give a good overview of the dynamically developing, complex field of knowledge to students. This monograph is highly recommended," BERND PLATZER, ZAAM In the present second volume the methods for describing the mechanical interactions in multiphase dynamics are provided. This fourth edition includes various updates, extensions, improvements and corrections. "The literature in the field of multiphase flows is numerous. Therefore, it is very important to have a comprehensive and systematic overview including useful numerical methods. The volumes have the character of a handbook and accomplish this function excellently. The models are described in detail and a great number of comprehensive examples and some cases useful for testing numerical solutions are included. These two volumes are very useful for scientists and practicing engineers in the fields of technical thermodynamics, chemical engineering, fluid mechanics, and for mathematicians with interest in technical problems. Besides, they can give a good overview of the dynamically developing, complex field of knowledge to students. This monograph is highly recommended," BERND PLATZER, ZAAM "The literature in the field of multiphase flows is numerous. Therefore, it is very important to have a comprehensive and systematic overview including useful numerical methods. The volumes have the character of a handbook and accomplish this function excellently. The models are described in detail and a great number of comprehensive examples and some cases useful for testing numerical solutions are included. These two volumes are very useful for scientists and practicing engineers in the fields of technical thermodynamics, chemical engineering, fluid mechanics, and for mathematicians with interest in technical problems. Besides, they can give a good overview of the dynamically developing, complex field of knowledge to students. This monograph is highly recommended," BERND PLATZER, ZAAM

The present Volume 4 of the successful monograh package "Multiphase Flow Dynamics"is devoted to selected Chapters of the multiphase fluid dynamics that are important for practical applications but did not find place in the previous volumes. The state of the art of the turbulence modeling in multiphase flows is presented. As introduction, some basics of the single phase boundary layer theory including some important scales and flow oscillation characteristics in pipes and rod bundles are presented. Then the scales characterizing the dispersed flow systems are presented. The description of the turbulence is provided at different level of complexity: simple algebraic models for eddy viscosity, simple algebraic models based on the Boussinesq hypothesis, modification of the boundary layer share due to modification of the bulk turbulence, modification of the boundary layer share due to nucleate boiling. The role of the following forces on the mathematical description of turbulent flows is discussed: the lift force, the lubrication force in the wall boundary layer, and the dispersion force. A pragmatic generalization of the k-eps models for continuous velocity field is proposed containing flows in large volumes and flows in porous structures. A Methods of how to derive source and sinks terms for multiphase k-eps models is presented. A set of 13 single- and two phase benchmarks for verification of k-eps models in system computer codes are provided and reproduced with the IVA computer code as an example of the application of the theory. This methodology is intended to help other engineers and scientists to introduce this technology step-by-step in their own engineering practice. In many practical application gases are solved in liquids under given conditions, released under other conditions and therefore affecting technical processes for good of for bad. Useful information on the solubility of oxygen, nitrogen, hydrogen and carbon dioxide in water under large interval of pressures and temperatures is collected, and appropriate mathematical approximation functions are provided. In addition methods for the computation of the diffusion coefficients are described. With this information solution and dissolution dynamics in multiphase fluid flows can be analyzed. For this purpose the non-equilibrium absorption and release on bubble, droplet and film surfaces under different conditions is mathematically described. A systematic set of internally consistent state equations for diesel fuel gas and liquid valid in broad range of changing pressure and temperature is provided. This new second edition includes various updates, extensions, improvements and corrections. In many practical application gases are solved in liquids under given conditions, released under other conditions and therefore affecting technical processes for good of for bad. Useful information on the solubility of oxygen, nitrogen, hydrogen and carbon dioxide in water under large interval of pressures and temperatures is collected, and appropriate mathematical approximation functions are provided. In addition methods for the computation of the diffusion coefficients are described. With this information solution and dissolution dynamics in multiphase fluid flows can be analyzed. For this purpose the non-equilibrium absorption and release on bubble, droplet and film surfaces under different conditions is mathematically described. A systematic set of internally consistent state equations for diesel fuel gas and liquid valid in broad range of changing pressure and temperature is provided. This new second edition includes various updates, extensions, improvements and corrections.

th This volume contains the papers presented at the 16 DGLR/STAB-Symposium held at the Eurogress Aachen and organized by RWTH Aachen University, Germany, November, 3 - 4, 2008. STAB is the German Aerospace Aerodynamics Association, founded towards the end of the 1970'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 discipline "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 problems. 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. Since 1986 the symposium has taken place at different locations in Germany every two years. In between STAB workshops regularly take place at the DLR in Goettingen.

The standard textbooks on aerodynamics usually omit any discussion of un steady aerodynamics or, at most, consider it only in a single chapter, based on two justifications. The first is that unsteady aerodynamics should be regarded as a specialized subject required "only" in connection with understanding and an alyzing aeroelastic phenomena such as flutter and gust response, and therefore should be dealt with in related specialist books. The second reason appears to be reluctance to discuss aerodynamics with the inclusion of the time-dependent terms in the conservation equations and the boundary conditions for fear that added complications may discourage the reader. We take the opposite view in this book and argue that a full understanding of the physics of lift generation is possible only by considering the unsteady aerody namics of the starting vortex generation process. Furthermore, certain "steady" flows are inherently unsteady in the presence of flow separation, as for example the unsteady flow caused by the Karman vortex shedding downstream of a cylin der and "static" airfoil stall which is an inherently unsteady flow phenomenon. Therefore, it stands to reason that a unified treatment of aerodynamics that yields steady-state aerodynamics as a special case offers advantages. This rea soning is strengthened by the developments in computational fluid dynamics over the past forty years, which showed that accurate steady-state solutions can be obtained efficiently by solving the unsteady flow equations.

This book explores the interplay of bubble dynamics and shock waves, covering shock wave emission by laser generated bubbles, pulsating bubbles near boundaries, interaction of shock waves with bubble clouds, applications in shock wave lithotripsy, and more.

This newly-translated book takes the reader from the basic principles and conservation laws of hydrodynamics to the description of general atmospheric circulation. Among the topics covered are the Kelvin, Ertel and Rossby-Obukhov invariants, quasi-geostrophic equation, thermal wind, singular Helmholtz vortices, derivation of the Navier-Stokes equation, Kolmogorov's flow, hydrodynamic stability, and geophysical boundary layers. Generalizing V. Arnold's approach to hydrodynamics, the author ingeniously brings in an analogy of Coriolis forces acting on fluid with motion of the Euler heavy top and shows how this is used in the analysis of general atmospheric circulation. This book is based on popular graduate and undergraduate courses given by F.V.Dolzhansky at the Moscow Institute of Physics and Technology, and is the result of the author's highly acclaimed work in Moscow's Laboratory of Geophysical Hydrodynamics. Each chapter is full of examples and figures, exercises and hints, motivating and illustrating both theoretical and experimental results. The exposition is comprehensive yet user-friendly in engaging and exploring the broad range of topics for students and researchers in mathematics, physics, meteorology and engineering.

The book provides personal memories along with description of scientific works written by ex-graduate students and research associates of the late Professor Glass. The described research work covers a wide range of shock wave phenomena, resulting from seeds planted by Professor Glass. Professor Glass was born in Poland in 1918. He immigrated together with his parents to Canada at the age of 12 and received all his professional education at the University of Toronto, Canada. He became a world recognized expert in shock wave phenomena, and during his 45 years of active research he supervised more than 125 master and doctoral students, post-doctoral fellows and visiting research associates. In this book seven of his past students/research-associates describe their personal memories of Professor Glass and present some of their investigations in shock wave phenomena which sprung from their past work with Professor Glass. Specifically, these investigations include underwater shock waves, shock/bubble interaction, medical applications of shock wave, various types of shock tubes and shock tube techniques, shock wave attenuation and different types of shock wave reflections.

I used the opportunity of this edition to correct some minor mistakes and clarify, wherever it possible, exposition of the theory in comparison with the previous edition of this book (Kluwer, Dordrechtet cet, 2000). It provokes - largement of the book, though I tried to present the modern theory of thermic motion of long macromolecules in compact form. I have tried to accumulate the common heritage and to take into account di?erent approaches in the theory of dynamics of linear polymers, at least, to understand and make clear the importance of various ideas for explanation of relaxation phenomena in linear polymers, to present recent development in the ?eld. The theory of non-equilibrium phenomena in polymer systems is based on the fundamental principles of statistical physics. However, the peculiarities of thestructureandthebehaviourofthesystemsnecessitatetheimplementation of special methods and heuristic models that are di?erent from those for gases and solids, so that polymer dynamics has appeared to be a special branch of physicsnow. Themonographcontainsdiscussionsofthemainprinciplesofthe theoryof slowrelaxationphenomena in linearpolymers, elaborated inthe last decades. The basic model of a macromolecule, which allows us a consistent explanation of di?erent relaxation phenomena (di?usion, neutron scattering, viscoelasticity, optical birefringence), remains to be a coarse-grained or be- spring model, considered in di?erent environments: viscous, to describe the behaviourofdilutesolutions,orviscoelastic,todescribethebehaviourofboth weakly and strongly entangled systems.

The book contains invited lectures and selected contributions presented at the Enzo Levi and XVII Annual Meeting of the Fluid Dynamic Division of the Mexican Physical Society in 2011. It is aimed to fourth year undergraduate and graduate students, and scientists in the field of physics, engineering and chemistry that have interest in Fluid Dynamics from the experimental and theoretical point of view. The invited lectures are introductory and avoid the use of complicate mathematics. The other selected contributions are also adequate to fourth year undergraduate and graduate students. The Fluid Dynamics applications include multiphase flow, convection, diffusion, heat transfer, rheology, granular material, viscous flow, porous media flow, geophysics and astrophysics. The material contained in the book includes recent advances in experimental and theoretical fluid dynamics and is adequate for both teaching and research.

This immensely practical guide to PIV provides a condensed, yet exhaustive guide to most of the information needed for experiments employing the technique. This second edition has updated chapters on the principles and extra information on microscopic, high-speed and three component measurements as well as a description of advanced evaluation techniques. What's more, the huge increase in the range of possible applications has been taken into account as the chapter describing these applications of the PIV technique has been expanded.

This volume comprises the communications presented at the ETC 11, the EUROMECH European Turbulence conference held in 2007 in Porto.

The scientific committee has chosen the contributions out of the following topics: Acoustics of turbulent flows; Atmospheric turbulence; Control of turbulent flows; Geophysical and astrophysical turbulence; Instability and transition; Intermittency and scaling; Large eddy simulation and related techniques; MHD turbulence; Reacting and compressible turbulence; Transport and mixing; Turbulence in multiphase and non-Newtonian flows; Vortex dynamics and structure formation; Wall bounded flows.

The "Turbulence and Interactions 2006" (TI2006) conference was held on the island of Porquerolles, France, May 29-June 2, 2006. The scientific sponsors of the conference were * Association Francaise de Mecanique, * CD-adapco, * DGA * Ecole Polytechnique Federale de Lausanne (EPFL), * ERCOFTAC : European Research Community on Flow, Turbulence and Combustion, * FLUENT, * The French Ministery of Foreign Affairs, * Laboratoire de Modelisation en Mecanique, Paris 6, * ONERA. The conference was a unique event. Never before have so many organisations concerned with turbulence works come together in one conference. As the title "Turbulence and Interactions" anticipated, the workshop was not run with parallel sessions but instead of one united gathering where people had strong interactions and discussions. Many of the 85 or so attendants were veterans of previous ERCOFTAC conferences. Some young researchers attended their very first int- national meeting. The organisers were fortunate in obtaining the presence of the following - vited speakers: N. Adams (TUM, Germany), C. Cambon (ECL, France), J.-P. Dussauge (Polytech Marseille, France), D.A. Gosman (Imperial College, UK), Y. Kaneda (Nagoya University, Japan), O. Simonin (IMFT, France), G. Tryggvason (WPI, USA), D. Veynante (ECP, France), F. Waleffe (University of Wisconsin, USA), Y.K. Zhou (University of California, USA). The topics covered by the 59 papers ranged from experimental results through theory to computations. The papers of the conference went through the usual - viewing process for two special issues of international journals : Computers and Fluids, and Flow, Turbulence and Combustion.

This volume contains 27 contributions to the Forth Russian-German Advanced Research Workshop on Computational Science and High Performance Computing presented in October 2009 in Freiburg, Germany. The workshop was organized jointly by the High Performance Computing Center Stuttgart (HLRS), the Institute of Computational Technologies of the Siberian Branch of the Russian Academy of Sciences (ICT SB RAS) and the Section of Applied Mathematics of the University of Freiburg (IAM Freiburg) The contributions range from computer science, mathematics and high performance computing to applications in mechanical and aerospace engineering. They show a wealth of theoretical work and simulation experience with a potential of bringing together theoretical mathematical modelling and usage of high performance computing systems presenting the state of the art of computational technologies.

Turbulence pervades our world, from weather patterns to the air entering our lungs. This book describes methods that reveal its structures and dynamics. Building on the existence of coherent structures - recurrent patterns - in turbulent flows, it describes mathematical methods that reduce the governing (Navier-Stokes) equations to simpler forms that can be understood more easily. This second edition contains a new chapter on the balanced proper orthogonal decomposition: a method derived from control theory that is especially useful for flows equipped with sensors and actuators. It also reviews relevant work carried out since 1995. The book is ideal for engineering, physical science and mathematics researchers working in fluid dynamics and other areas in which coherent patterns emerge.

The second, enlarged edition of this established reference integrates many new insights into wastewater hydraulics. This work serves as a reference for researchers but also is a basis for practicing engineers. It can be used as a text book for graduate students, although it has the characteristics of a reference book. It addresses mainly the sewer hydraulician but also general hydraulic engineers who have to tackle many a problem in daily life, and who will not always find an appropriate solution. Each chapter is introduced with a summary to outline the contents. To illustrate application of the theory, examples are presented to explain the computational procedures. Further, to relate present knowledge to the history of hydraulics, some key dates on noteworthy hydraulicians are quoted. A historical note on the development of wastewater hydraulics is also added. References are given at the end of each chapter, and they are often helpful starting points for further reading. Each notation is defined when introduced, and listed alphabetically at the end of each chapter. This new edition includes in particular sideweirs with throttling pipes, drop shafts with an account on the two-phase flow features, as well as conduit choking due to direct or undular hydraulic jumps.