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

In Large Eddy Simulation of turbulent ∞ows, the Navier{Stokes equations are con- volved with a fllter and difierentiation 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 Lp(Rd) (i.e., it vanishes as the averaging radius - ! 0) if and only if the ∞uid and the boundary exert exactly zero force on each other. Next, we show that the commutation error tends to zero in Hi1(›) as - ! 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 in∞uence of the commutation error on the energy balance of the flltered equations.