Volumes 1-3 [36] give a modern version of calculus and linear algebra including computation starting at a basic undergraduate level, and subsequent volumes on a graduate level cover di?erent areas of applications with focus on computational methods: • Volume 4: Computational Turbulent Incompressible Flow.

Overview.- Main Objective.- Mysteries and Secrets.- Turbulent Flow and History of Aviation.- The Euler Equations.- The Incompressible Euler and Navier–Stokes Equations.- Triumph and Failure of Mathematics.- Laminar and Turbulent Flow.- Computational Turbulence.- A First Study of Stability.- d'Alembert's Mystery and Bernoulli's Law.- Prandtl's Resolution of d'Alembert's Mystery.- New Resolution of d'Alembert's Mystery.- Mathematics of Turbulence.- Turbulence and Chaos.- A $1 Million Prize Problem.- Weak Uniqueness by Computation.- Existence of ?-Weak Solutions by G2.- Stability Aspects of Turbulence in Model Problems.- A Convection-Diffusion Model Problem.- G2 for Euler.- Summary of Mathematical Aspects.- Secrets.- Secrets of Ball Sports.- Secrets of Flight.- Secrets of Sailing.- Secrets of Racing.- Computational Method.- Reynolds Stresses In and Out.- Smagorinsky Viscosity In and Out.- Friction Boundary Condition as Wall Model.- G2 for Navier-Stokes Equations.- A Discrete Solver.- G2 as Adaptive DNS/LES.- Implementation of G2 with FEniCS.- Moving Meshes and ALE Methods.- Flow Fundamentals.- Bluff Body Flow.- Boundary Layers.- Separation.- Transition to Turbulence.- Loschmidt's Mystery.- Thermodynamics.- Joule's 1845 Experiment.- Compressible Euler in 1d.- Burgers' Equation.- Compressible Euler in 3d.- EG2 for Compressible Flow.- Philosophy of EG2.- Does God Really Play Dice?.

This is Volume 4 of the book series of the Body & Soul mathematics education reform program, and presents a unified new approach to computational simulation of turbulentflow starting from the general basis of calculus and linear algebra of Vol 1-3. The book puts the Body & 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. The book shows that direct application of G2 without any turbulence or wall modeling, allows reliable computation on a PC of mean value quantities of turbulent flow such as drag and lift.The power of G2 is demonstrated by resolving several classical scientific paradoxes of fluid flow and by uncovering secrets of flying, sailing, racing and ball sports. The book presents new aspects on both mathematics and computation of turbulent flow, and challenges established approaches. The book is directed to a wide audience of computational mathematicians fluid dynamicists and scientists. The G2 solver is available as part the free software project FEniCS at www.fenics.org. The book has a dedicated dynamic web page, including movies from a wide variety of simulations,  at www.bodysoulmath.org. The book is focussed on incompressible flow, but opens to compressible flow continued in Vol 5 on thermodynamics. The authors are experts on computational mathematics and technology.

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