### 2002-01

#### The Commutation Error of the Space Averaged Navier-Stokes Equations on a Bounded Domain

Series: 2002-01, Preprints

MSC:
35Q30 ~Navier-Stokes equations

Abstract:
In Large Eddy Simulation
of turbulent flows, the Navier--Stokes equations are convolved with
a filter and differentiation and convolution are interchanged,
introducing
an extra commutation error term, which is nearly
universally dropped from the resulting equations. We show that the
commutation error
is asymptotically negligible in $L^p(\mathbb R^d)$
(i.e., it vanishes as the averaging
radius $\delta \to 0$) if and only if the fluid and the boundary
exert exactly zero force on each other. Next, we show that
the commutation error tends to zero in $H^{-1}(\Omega)$
as $\delta\to 0$. Convergence is proven also for a weak form of the
commutation error. The order of convergence is studied in both cases.
Last, we study the
influence of the commutation error on the energy balance of the
filtered equations.

Keywords:
large eddy simulation, commutation error