# Computational Turbulent Incompressible Flow

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 mathematics education reform project Applied Mathematics Body and Soul. Here you can find a variety of material complementing and expanding the book, such as computational software including output in the form of movies, educational material, presentations, 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 without convincing anwers into our time. The reason is that slightly viscous flow always is turbulent and defies analytical mathematical representation. Accordingly, one of the seven Clay Mathematical Institute \$1 million Millenium Prize problems concerns existence, uniqueness and regularity of solutions to the Navier-Stokes equations.

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, referred to as General Galerkin or G2. The book presents a variety of new results on turbulent flow challenging commonly used techniques for computational simulation and modeling of turbulence including Reynolds Averaged Navier-Stokes RANS, and Large Eddy Simulation LES with Smagorinsky turbulent viscosity. The power of G2 is demonstrated in new solutions to classical open problems or Mysteries 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,
• The Clay Mathematics Institute Millenium Prize Problem on Navier-Stokes equations,

and a variety of advanced applications unraveling Secrets of

• flying,
• sailing,
• racing,
• ball sports.

Further, the Reception by different scientific communities of the new ideas and results of the book is presented on a debate forum including referee's reviews.