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 turbulent flow 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.