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By Johan Hoffman

This can be quantity four of the booklet sequence of the physique and Soul arithmetic schooling reform application. It provides a unified new method of computational simulation of turbulent circulate ranging from the final foundation of calculus and linear algebra of Vol 1-3. The publication places the physique and Soul computational finite point technique within the kind of basic Galerkin (G2) up opposed to the problem of computing turbulent ideas of the inviscid Euler equations and the Navier-Stokes equations with small viscosity. this is often a great textbook providing lots of new fabric with a good pedagogical procedure.

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This is still today the only analytical mathematical existence result for the general form of the NS equations! Leray did not prove any result on uniqueness of weak solutions of NS equations. Since uniqueness directly couples to stability, this means that effectively Leray did not consider the aspect of stability and in particular not the question of output uniqueness or weak uniqueness, which for turbulent flows has to replace pointwise uniqueness: We can estimate the difference in output of two approximate weak solutions by S if bounds the two residuals, so that the output will be unique up to S .

We shall see that by using slip or friction boundary conditions we avoid having to resolve all boundary layers, while with adaptivity crucial details such as separation may be captured without an excessive number of mesh points. Effectively, this means that we may reduce from the impossible > 1016 to possible < 108 number of mesh points for a complete aircraft. The reduction from impossible to possible is opened by G2 adaptivity with automatic turbulence modeling including slip/friction boundary conditions.

Using the NS equations to make predictions, it may be natural to speak of a total prediction error in output including errors from both data and computation. Usually however, we connect predictability to output error from data, and computability to output error from computation. We have indicated that in turbulent flow pointwise values in space-time of velocity-pressure are not predictable/computable to any tolerance of interest. The main question of this book thus concerns the predictability and computability of mean values in space-time of turbulent flow up to tolerances of interest.

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