02-01
The Commutation Error of the Space Averaged Navier-Stokes Equations on a Bounded Domain
by Dunca, A.; John V.; Layton W.J.
Preprint series: 02-01 , Preprints
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
Notes: additional MSC2000: 76F65
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