Mathematics of Two-dimensional Turbulence: Solutions to some exercises

Mathematics of Two-dimensional Turbulence: Solutions to some exercises PDF

Author: Sergej B. Kuksin

Publisher:

Published: 2012

Total Pages:

ISBN-13: 9781139570091

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"This book deals with basic problems and questions, interesting for physicists and engineers working in the theory of turbulence. Accordingly Chapters 3-5 (which form the main part of this book) end with sections, where we explain the physical relevance of the obtained results. These sections also provide brief summaries of the corresponding chapters. In Chapters 3 and 4, our main goal is to justify, for the 2D case, the statistical properties of fluid's velocity"--

Mathematics of Two-Dimensional Turbulence

Mathematics of Two-Dimensional Turbulence PDF

Author: Sergei Kuksin

Publisher: Cambridge University Press

Published: 2012-09-20

Total Pages: 337

ISBN-13: 113957695X

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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.

Mathematics of Two-dimensional Turbulence: Miscellanies

Mathematics of Two-dimensional Turbulence: Miscellanies PDF

Author: Sergej B. Kuksin

Publisher:

Published: 2012

Total Pages:

ISBN-13: 9781139579575

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"This book deals with basic problems and questions, interesting for physicists and engineers working in the theory of turbulence. Accordingly Chapters 3-5 (which form the main part of this book) end with sections, where we explain the physical relevance of the obtained results. These sections also provide brief summaries of the corresponding chapters. In Chapters 3 and 4, our main goal is to justify, for the 2D case, the statistical properties of fluid's velocity"--

Randomly Forced Nonlinear PDEs and Statistical Hydrodynamics in 2 Space Dimensions

Randomly Forced Nonlinear PDEs and Statistical Hydrodynamics in 2 Space Dimensions PDF

Author: Sergej B. Kuksin

Publisher: European Mathematical Society

Published: 2006

Total Pages: 108

ISBN-13: 9783037190210

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This book gives an account of recent achievements in the mathematical theory of two-dimensional turbulence, described by the 2D Navier-Stokes equation, perturbed by a random force. The main results presented here were obtained during the last five to ten years and, up to now, have been available only in papers in the primary literature. Their summary and synthesis here, beginning with some preliminaries on partial differential equations and stochastics, make this book a self-contained account that will appeal to readers with a general background in analysis. After laying the groundwork, the author goes on to recent results on ergodicity of random dynamical systems, which the randomly forced Navier-Stokes equation defines in the function space of divergence-free vector fields, including a Central Limit Theorem. The physical meaning of these results is discussed as well as their relations with the theory of attractors. Next, the author studies the behaviour of solutions when the viscosity goes to zero. In the final section these dynamical methods are used to derive the so-called balance relations--the infinitely many algebraical relations satisfied by the solutions.

Mathematics of Two-dimensional Turbulence: Appendix

Mathematics of Two-dimensional Turbulence: Appendix PDF

Author: Sergej B. Kuksin

Publisher:

Published: 2012

Total Pages:

ISBN-13: 9781139573528

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"This book deals with basic problems and questions, interesting for physicists and engineers working in the theory of turbulence. Accordingly Chapters 3-5 (which form the main part of this book) end with sections, where we explain the physical relevance of the obtained results. These sections also provide brief summaries of the corresponding chapters. In Chapters 3 and 4, our main goal is to justify, for the 2D case, the statistical properties of fluid's velocity"--

Numerical Studies in Two-dimensional Turbulence

Numerical Studies in Two-dimensional Turbulence PDF

Author: Fayeza Salim Sulti

Publisher:

Published: 2012

Total Pages:

ISBN-13:

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Two-dimensional turbulence has been extensively studied over the past years theoretically and numerically since the theory of the dual cascade energy. Numerical studies have revealed an impor- tant feature of two-dimensional turbulence, that is, the predomi- nance of coherent structures, followed by interaction and merger of these isolated vortices in the subsequent evolution. A method of 'vortex census' has been introduced to keep track of the vortices but the relation to reconnection has remained unexplored. In this Thesis, we study the reconnection process of vorticity con- tours associated with coherent vortices in two-dimensional turbu- lence for different Reynolds number. After checking topological integrity by the Euler index theorem, we make use of the critical points and their connectivity (so-called surface networks) to study the topological changes of vorticity contours. Wc show how this method can remarkably distinguish the dynamics of the vortic- ity field in the Navier-Stokes equations and that of the Charney- Hasegawa-Mima equation. We found that the potential vorticity formed vortex crystals. This excites us to study the vortex crystal in details by study a coarse-grained asymptotic equation [Smirnov and Chukbar(2001)]. Self-similar blow-up solutions with an infi- 1 I i :. nite total energy were given. We ask whether or not finite-time blow-up can take place developing from smooth initial data with a finite energy.

Modeling in Fluid Mechanics

Modeling in Fluid Mechanics PDF

Author: Igor Gaissinski

Publisher: CRC Press

Published: 2018-06-13

Total Pages: 534

ISBN-13: 1351029045

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This volume is dedicated to modeling in fluid mechanics and is divided into four chapters, which contain a significant number of useful exercises with solutions. The authors provide relatively complete references on relevant topics in the bibliography at the end of each chapter.

Supplementary Material and Solutions Manual for Mathematical Modeling in the Environment

Supplementary Material and Solutions Manual for Mathematical Modeling in the Environment PDF

Author: Charles R. Hadlock

Publisher: American Mathematical Soc.

Published: 2020-05-05

Total Pages: 199

ISBN-13: 0883857138

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This manual is meant to provide supplementary material and solutions to the exercises used in Charles Hadlock's textbook, Mathematical Modeling in the Environment. The manual is invaluable to users of the textbook as it contains complete solutions and often further discussion of essentially every exercise the author presents in his book. This includes both the mathematical/computational exercises as well as the research questions and investigations. Since the exercises in the textbook are very rich in content, (rather than simple mechanical problems), and cover a wide range, most readers will not have the time to work out every one on their own. Readers can thus still benefit greatly from perusing solutions to problems they have at least thought about briefly. Students using this manual still need to work out solutions to research questions using their own sources and adapting them to their own geographic locations, or to numerical problems using their own computational schemes, so this manual will be a useful guide to students in many course contexts. Enrichment material is included on the topics of some of the exercises. Advice for teachers who lack previous environmental experience but who want to teach this material is also provided and makes it practical for such persons to offer a course based on these volumes. This book is the essential companion to Mathematical Modeling in the Environment.

Fluid Dynamics

Fluid Dynamics PDF

Author: Z.U.A. Warsi

Publisher: CRC Press

Published: 2005-07-26

Total Pages: 874

ISBN-13: 0849333970

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Many introductions to fluid dynamics offer an illustrative approach that demonstrates some aspects of fluid behavior, but often leave you without the tools necessary to confront new problems. For more than a decade, Fluid Dynamics: Theoretical and Computational Approaches has supplied these missing tools with a constructive approach that made the book a bestseller. Now in its third edition, it supplies even more computational skills in addition to a solid foundation in theory. After laying the groundwork in theoretical fluid dynamics, independent of any particular coordinate system in order to allow coordinate transformation of the equations, the author turns to the technique of writing Navier–Stokes and Euler’s equations, flow of inviscid fluids, laminar viscous flow, and turbulent flow. He also includes requisite mathematics in several “Mathematical Expositions” at the end of the book and provides abundant end-of-chapter problems. What’s New in the Third Edition? New section on free surface flow New section on instability of flows through Chaos and nonlinear dissipative systems New section on formulation of the large eddy simulation (LES) problem New example problems and exercises that reflect new and important topics of current interest By integrating a strong theoretical foundation with practical computational tools, Fluid Dynamics: Theoretical and Computational Approaches, Third Edition is an indispensable guide to the methods needed to solve new and unfamiliar problems in fluid dynamics.

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