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- Clay Mathematics Institute Millenium Prize Problem on Navier-Stokes equations,

- The Clay Mathematics Institute Millenium Prize Problem on Navier-Stokes equations,

- flying
- sailing
- racing

- flying,
- sailing,
- racing,

and and a variety of advanced applications unraveling Secrets of

and a variety of advanced applications unraveling Secrets of

for computational simulation and modeling of turbulence including Reynolds Averaged Navier-Stokes RANS,

for computational simulation and modeling of turbulence including Reynolds Averaged Navier-Stokes RANS, and

Welcome to the homepage of the book Computational Turbulent Incompressible Flow, Volume 4 in the Springer book series of the

Welcome to the homepage of the book Computational Turbulent Incompressible Flow, Volume 4 in the Springer book series Applied Mathematics: Body and Soul of the

Welcome to the homepage of the book [[http://www.springer.com/west/home/generic/search/results?SGWID=4-40109-22-173700518-0 | Computational Turbulent Incompressible Flow, Volume 4]] in the Springer book series of the

Welcome to the homepage of the book Computational Turbulent Incompressible Flow, Volume 4 in the Springer book series of the

Welcome to the homepage of the book Computational Turbulent Incompressible Flow, Volume 4 in the Springer book series of the

Welcome to the homepage of the book [[http://www.springer.com/west/home/generic/search/results?SGWID=4-40109-22-173700518-0 | Computational Turbulent Incompressible Flow, Volume 4]] in the Springer book series of the

%The homepage is organized into

%*Forum: Mysteries, Secrets and Reception of New Ideas %*Computations Expo %*Education %*Software - FEniCS

Further, the Reception by different scientific communities of the new ideas and results of the book,Fori

Further, the Reception by different scientific communities of the new ideas and results of the book

classical open problems or Mysteries including

classical open problems or Mysteries including

and and a variety of advanced applications unraveling Secrets of

and and a variety of advanced applications unraveling Secrets of

Further, the Reception by different scientific communities of the new ideas and results of the book,Fori

Further, the Reception by different scientific communities of the new ideas and results of the book,Fori

such as computational software including output in the form of movies, educational material,

such as computational software including output in the form of movies, educational material,

and a forum for debate of key issues.

Further, the Reception by different scientific communities of the new ideas and results of the book,

The homepage is organized into

%The homepage is organized into

- Forum: Mysteries, Secrets and Reception of New Ideas
- Computations Expo
- Education
- Software - FEniCS

%*Forum: Mysteries, Secrets and Reception of New Ideas %*Computations Expo %*Education %*Software - FEniCS

such as computational software including output in the form of movies, educational material,

such as computational software including output in the form of movies, educational material,

is presented in referee's reviews.

is presented on a debate forum including referee's reviews.

Further, the Reception by different scientific communities of the new results of the book,

Further, the Reception by different scientific communities of the new ideas and results of the book,

Further, the Reception by different scientific of the new results presented

Further, the Reception by different scientific communities of the new results of the book,

- Clay Mathematics Institute Millenium Prize Problem on Navier-Stokes equations,

Further, the official formulation of the Clay Institute Prize Problem is challenged and a new formulation and (partial) resolution is presented.

Further, the Reception by different scientific of the new results presented is presented in referee's reviews.

classical open problems including

classical open problems or Mysteries including

and and a variety of advanced applications unraveling secrets of

and and a variety of advanced applications unraveling Secrets of

The homepage is organized into

Large Eddy Simulation LES and Smagorinsky turbulent viscosity. The power of G2 is demonstrated in new solutions to

Large Eddy Simulation LES with Smagorinsky turbulent viscosity. The power of G2 is demonstrated in new solutions to

the Navier-Stokes equations.

existence, uniqueness and regularity of solutions to the Navier-Stokes equations.

turbulent and defies analytical mathematical representation. One of the seven Clay Mathematical Institute $1 million Millenium Prize problems concerns

turbulent and defies analytical mathematical representation. Accordingly, one of the seven Clay Mathematical Institute $1 million Millenium Prize problems concerns

such as computational software including output in the form of movies

such as computational software including output in the form of movies, educational material, presentations

Welcome to the homepage of Computational Turbulent Incompressible

Welcome to the homepage of the book Computational Turbulent Incompressible

- Loschmidt's paradox of irreversibility in reversible Hamiltonian systems.

- Loschmidt's paradox of irreversibility in reversible Hamiltonian systems,

Further, the official formulation of the Clay Institute Prize Problem is challenged.

Further, the official formulation of the Clay Institute Prize Problem is challenged and a new formulation and (partial) resolution is presented.

The book presents a fresh attack on the open problems based on finite element computation and a new concept of output uniqueness combined into duality-based a posteriori error estimation

The book presents a fresh attack on the open problems of turbulent flow, based on finite element computation and a new concept of output uniqueness combined with duality-based a posteriori error estimation,

of turbulence modeling including Reynolds Averaged Navier-Stokes RANS, Large Eddy Simulation LES and Smagorinsky viscosity. The power of G2 is demonstrated in new solutions to

for computational simulation and modeling of turbulence including Reynolds Averaged Navier-Stokes RANS, Large Eddy Simulation LES and Smagorinsky turbulent viscosity. The power of G2 is demonstrated in new solutions to

these open problems.

the Navier-Stokes equations.

turbulent, which defies analytical mathematical representation. Accordingly one of the seven Clay Mathematical Institute $1 million Millenium Prize problems concerns

turbulent and defies analytical mathematical representation. One of the seven Clay Mathematical Institute $1 million Millenium Prize problems concerns

solutions have remained without convincing anwers into our time. The reason is that slightly viscous flow always is

solutions, have remained without convincing anwers into our time. The reason is that slightly viscous flow always is

Here you can find a variety of material complementing and expanding the material presented in the book such as computational software including output in the form of movies

Here you can find a variety of material complementing and expanding the book, such as computational software including output in the form of movies

solutions have remained essentially open into our time. The reason is that slightly viscous flow always is turbulent.

solutions have remained without convincing anwers into our time. The reason is that slightly viscous flow always is turbulent, which defies analytical mathematical representation.

the open questions.

The book presents a fresh attack on the basic open problems based on finite element

these open problems.

The book presents a fresh attack on the open problems based on finite element

of turbulence modeling including Reynolds Averaged Navier-Stokes RANS, Large Eddy Simulation LES, Smagorinsky turbulent viscosity. The power of G2 is demonstrated in new solutions to

of turbulence modeling including Reynolds Averaged Navier-Stokes RANS, Large Eddy Simulation LES and Smagorinsky viscosity. The power of G2 is demonstrated in new solutions to

The web page contains a Forum for exchange of ideas, and Expo for computational results, aspects of Education and software:

The book presents a fresh attack on the basic problems based on finite element

The book presents a fresh attack on the basic open problems based on finite element

Smagorinsky turbulent viscosity. The power of G2 is demonsrted in new solutions to classical open problems including d'Alembert's paradox of zero drag in inviscid flow, Sommerfeld's paradox of stability of Couette flow at all Reynolds numbers, and Loschmidt's paradox of irreversibility in reversible Hamiltonian systems.

Smagorinsky turbulent viscosity. The power of G2 is demonstrated in new solutions to classical open problems including

- d'Alembert's paradox of zero drag in inviscid flow,
- Sommerfeld's paradox of stability of Couette flow at all Reynolds numbers,
- Loschmidt's paradox of irreversibility in reversible Hamiltonian systems.

and and a variety of advanced applications unraveling secrets of

- flying
- sailing
- racing
- ball sports.

The material in this book in many ways goes against what is traditionally found in most text books of mathematical fluid dynamics. Since our approach is new, and may also be considered controversial by some, we here want to open for a free debate on these very important questions that are fundamental for both mathematics and fluid dynamics. At the Forum of this homepage we hope for lively discussions on these important questions.

The web page contains a Forum for exchange of ideas, and Expo for computational results, aspects of Education and software:

Accordingly one of the seven Clay Mathematical Institute $1 million Millenium Prize problems concerns these precisely these questions.

The book presents a fresh attack on the basic problems based on a combination of finite precision computation and a new concept of output uniqueness materialized in duality-based a posteriori error estimation in stabilized Galerkin finite element methods referred to as General Galerkin or G2. The book presents a variety of new results challenging commonly accepted views. In particular, (new) solutions to d'Alembert's paradox of zero drag, Sommerfeld's paradox of stability of Couette flow nad Loschmidt's paradox of irreversibility in reversible Hamiltonian systems, are presented, together with a critical analysis of the offcial formulation of the Clay Institute Prize Problem. ies; such as the d'Alembert mystery of drag in an inviscid flow, the Sommerfeld mystery of transition to turbulence in shear flow, and the Loschmidt mystery of violation of the 2nd law of thermodynamics. We approach these mysteries using a synthesis of computational and analytical mathematics, and in the book we offer new solutions.

Accordingly one of the seven Clay Mathematical Institute $1 million Millenium Prize problems concerns the open questions.

The book presents a fresh attack on the basic problems based on finite element computation and a new concept of output uniqueness combined into duality-based a posteriori error estimation referred to as General Galerkin or G2. The book presents a variety of new results on turbulent flow challenging commonly used techniques of turbulence modeling including Reynolds Averaged Navier-Stokes RANS, Large Eddy Simulation LES, Smagorinsky turbulent viscosity. The power of G2 is demonsrted in new solutions to classical open problems including d'Alembert's paradox of zero drag in inviscid flow, Sommerfeld's paradox of stability of Couette flow at all Reynolds numbers, and Loschmidt's paradox of irreversibility in reversible Hamiltonian systems. Further, the official formulation of the Clay Institute Prize Problem is challenged.

solutions have remained essentially open into our time. Accordingly one of the seven

solutions have remained essentially open into our time. The reason is that slightly viscous flow always is turbulent. Accordingly one of the seven

these questions.

these questions.

On the homepage you can find a variety of material complementing and expanding the book including computational software and movies and a forum for debate of key issues of the book.

The fundamental equations for fluid flow have been known for almost 200 years, but still today the basic mathematical question of whether there exists a solution to the equations has not been settled. In fact, this problem is one of the famous Clay Institute $1 million Prize problems. In this book we give our own critical view on the formulation of this Prize problem, and we propose a possible alternative formulation of the problem. We also address several other mysteries related to mathematical modeling of fluid flow that have haunted scientist over the centuries; such as the d'Alembert mystery of drag in

Here you can find a variety of material complementing and expanding the material presented in the book such as computational software including output in the form of movies and a forum for debate of key issues.

The basic mathematical models of fluid flow, the Euler and Navier-Stokes equations, have been known for almost 200 years, but the basic mathematical questions of existence, uniqueness and stability and the related basic question of the accuracy of computational solutions have remained essentially open into our time. Accordingly one of the seven Clay Mathematical Institute $1 million Millenium Prize problems concerns these precisely these questions.

The book presents a fresh attack on the basic problems based on a combination of finite precision computation and a new concept of output uniqueness materialized in duality-based a posteriori error estimation in stabilized Galerkin finite element methods referred to as General Galerkin or G2. The book presents a variety of new results challenging commonly accepted views. In particular, (new) solutions to d'Alembert's paradox of zero drag, Sommerfeld's paradox of stability of Couette flow nad Loschmidt's paradox of irreversibility in reversible Hamiltonian systems, are presented, together with a critical analysis of the offcial formulation of the Clay Institute Prize Problem. ies; such as the d'Alembert mystery of drag in

Welcome to the homepage of the new book Computational Turbulent Incompressible Flow, Volume 4 of the Springer book series of the

Welcome to the homepage of Computational Turbulent Incompressible Flow, Volume 4 in the Springer book series of the

On the homepage you can a variety of material complementing and expanding the book including computational results, software and a forum for debate of key issues of the book.

On the homepage you can find a variety of material complementing and expanding the book including computational software and movies and a forum for debate of key issues of the book.

Flow, which is also volume 4 in the Springer book series connected to the

Flow, Volume 4 of the Springer book series of the

Here you can find useful material to complement the book, as well as participate in the open debate around subjects from the book.

On the homepage you can a variety of material complementing and expanding the book including computational results, software and a forum for debate of key issues of the book.

mathematics and fluid dynamics. At the forum of this homepage

mathematics and fluid dynamics. At the Forum of this homepage

Although the fundamental equations for fluid flow have been known for almost 200 years, still today the basic mathematical question of whether there exists a solution

The fundamental equations for fluid flow have been known for almost 200 years, but still today the basic mathematical question of whether there exists a solution

Already Leonardo da Vinci sought to capture the basic features of turbulent flow in his sketches, and later in the 18th and 19th centuries the invention of calculus opened for a mathematical description of fluid flow in the form of the viscous Navier-Stokes equations, and the inviscid Euler equations. Although the fundamental equations have been known for almost 200 years, still today the basic mathematical questions of whether there exists a classical solution to the equations have not been settled. In fact, this problem is one of the famous

Although the fundamental equations for fluid flow have been known for almost 200 years, still today the basic mathematical question of whether there exists a solution to the equations has not been settled. In fact, this problem is one of the famous

that have haunted scientist over the centuries; such as the dAlembert mystery of drag in

that have haunted scientist over the centuries; such as the d'Alembert mystery of drag in

and in the book we offer new solutions to these mysteries that we believe is more to the point than the traditional views.

and in the book we offer new solutions.

mathematics and fluid dynamics. At the forum of this homepage we have a mailinglist which we hope will host lively discussions on these important questions.

mathematics and fluid dynamics. At the forum of this homepage we hope for lively discussions on these important questions.

- Software - fenics

- Software - FEniCS

in the open debate around subjects from the book by joining the maillinglist.

in the open debate around subjects from the book.

<a href="http://www.claymath.org/millennium/Navier-Stokes_Equations/">Clay Institute $1 million Prize problems</a>.

<br>

Clay Institute $1 million Prize problems.

<br>

mathematics and fluid dynamics. At <a href=wip.html>the forum of this homepage</a>

mathematics and fluid dynamics. At the forum of this homepage

</td> </tr>

Already Leonardo da Vinci sought to capture the basic features of turbulent flow in his sketches, and later in the 18th and 19th centuries the invention of calculus opened for a mathematical description of fluid flow in the form of the viscous Navier-Stokes equations, and the inviscid Euler equations. Although the fundamental equations have been known for almost 200 years, still today the basic mathematical questions of whether there exists a classical solution to the equations have not been settled. In fact, this problem is one of the famous <a href="http://www.claymath.org/millennium/Navier-Stokes_Equations/">Clay Institute $1 million Prize problems</a>.

<br>

In this book we give our own critical view on the formulation of this Prize problem, and we propose a possible alternative formulation of the problem.

We also address several other mysteries related to mathematical modeling of fluid flow that have haunted scientist over the centuries; such as the dAlembert mystery of drag in an inviscid flow, the Sommerfeld mystery of transition to turbulence in shear flow, and the Loschmidt mystery of violation of the 2nd law of thermodynamics.

<br>

We approach these mysteries using a synthesis of computational and analytical mathematics, and in the book we offer new solutions to these mysteries that we believe is more to the point than the traditional views.

The material in this book in many ways goes against what is traditionally found in most text books of mathematical fluid dynamics.

Since our approach is new, and may also be considered controversial by some, we here want to open for a free debate on these very important questions that are fundamental for both mathematics and fluid dynamics. At <a href=wip.html>the forum of this homepage</a> we have a mailinglist which we hope will host lively discussions on these important questions.

</td> </tr>

Computational Turbulent Incompressible Flow

# Computational Turbulent Incompressible Flow

# Welcome to FlowWiki!

Computational Turbulent Incompressible Flow

http://www.csc.kth.se/~jhoffman/graphics/coverv4-2.png

http://www.csc.kth.se/~jhoffman/graphics/coverv4-2.png

- Software - fenics

- Software - fenics

Welcome to the homepage of the new book Computational Turbulent Incompressible Flow, which is also volume 4 in the Springer book series connected to the mathematics education reform project Applied Mathematics Body and Soul.

Here you can find useful material to complement the book, as well as participate in the open debate around subjects from the book by joining the maillinglist.

- Software - fenics

- Software - fenics

http://www.csc.kth.se/~jhoffman/graphics/coverv4-2.png

http://www.csc.kth.se/~jhoffman/graphics/coverv4-2.png

http://www.csc.kth.se/~jhoffman/graphics/coverv4-2.png

http://www.csc.kth.se/~jhoffman/graphics/coverv4-2.png

http://www.csc.kth.se/~jhoffman/graphics/coverv4-2.png";

http://www.csc.kth.se/~jhoffman/graphics/coverv4-2.png

http://www.csc.kth.se/~jhoffman/graphics/coverv4-2.png";

http://www.csc.kth.se/~jhoffman/graphics/coverv4-2.png";

- Software - fenics

- Software - fenics

http://www.csc.kth.se/~jhoffman/graphics/coverv4-2.png";

- Software - fenics

- Software

- Software - fenics

- http://www.fenics.org

- Software

Welcome to FlowWiki!

# Welcome to FlowWiki!

- http://www.fenics.org

Welcome to PmWiki!

A local copy of PmWiki's documentation has been installed along with the software, and is available via the documentation index.

To continue setting up PmWiki, see initial setup tasks.

The basic editing page describes how to create pages in PmWiki. You can practice editing in the wiki sandbox.

More information about PmWiki is available from http://www.pmwiki.org .

Welcome to FlowWiki!