We develop a new method for averaging the Navier-Stokes equations. The
idea is to "fuzzy" the Lagrangian flow map f of the Navier-Stokes
equations;
the fuzzy flow takes a particle x to an a-radius neighborhood of the exact
position f(x). The fuzzy flow consists of near-identity volume-preserving
perturbations of the exact N-S flow map f. We create a new action
principle based on the new fuzzy flow map, asymptotically expand in the
fuzzying parameter a, and average over all possible near-identity
measure-preserving transformations. The first variation of this procedure
yields the LANS equations, a coupled system of PDEs of the pair (u,F),
where u is the mean velocity, and F is the covariance tensor. Unlike the
Reynolds Averaged Navier-Stokes (RANS), there is no closure problem, and
no ad hoc dissipation is added into the system.
We shall present the derivation, analytic results concerning
well-posedness (which is global in 3D), and numerical results for decaying
isotropic turbulence.
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