### 2002-29

#### Approximating local averages of fluid velocities: the equilibrium Navier--Stokes equations

Series: 2002-29, Preprints

MSC:
65N30 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods

Abstract:
In the approximation of higher Reynolds number flow problems, the usual
approach is to seek to approximate suitable velocity averages rather than
the pointwise fluid velocity itself. We consider an approach to this
question wherein the averages are local, spatial averages computed with
the Gaussian filter (as in large eddy simulation) and the averages
are approximated without using either turbulent closure models or
wall laws. The approach we consider is a (underresolved) direct
numerical simulation followed by postprocessing to extract accurate
flow averages. \'A priori and a posteriori
estimates are given for $\| g_\delta\ast(\bu-\bu^h)\|_0$
which can give guidance for the coupling between the averaging radius
$\delta$ and the mesh width $h$. Numerical experiments support the
error estimates and illustrate the adaptive grid refinement procedure.

Keywords:
large eddy simulation, postprocessing, convergence of the finite element method