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A simple procedure to approximate the noncommutation terms that arisewhenever it is necessary to use a variable scale filtering of the motionequations and to compensate directly the flow solutions from the commutationerror is here presented. Such a situation usually concerns large eddysimulation of nonhomogeneous turbulent flows. The noncommutation of the averageand differentiation operations leads to nonhomogeneous terms in the motionequations, that act as source terms of intensity which depend on the gradientof the filter scale and which, if neglected, induce a systematic errorthroughout the solution. Here the different noncommutation terms of the motionequation are determined as functions of the \delta-gradient and of the\delta-derivatives of the filtered variables. It is shown here thatapproximated noncommutation terms of the fourth order of accuracy, with respectto the filtering scale, can be obtained using series expansions in the filterwidth of approximations based on finite differences and introducing successivelevels of filtering, which makes it suitable to use in conjunction with dynamicor mixed subgrid models. The procedure operates in a way which is independentof the type of filter in use and without increasing the differential order ofthe equations, which, on the contrary, would require additional boundaryconditions. It is not necessary to introduce a mapping function of thenonuniform grid in the physical domain into a uniform grid in an infinitedomain. A priori}tests on the turbulent channel flow (Re_\tau=180 and 590)highlight the approximation capability of the present procedure. A numericalexample is given, which draws attention to the nonlocal effects on the solutiondue to the lack of noncommutation terms in the motion equation and to theefficiency of the present procedure in reducing the commutation error on thesolution.

Variable scale filtered Navier-Stokes Equations. A new procedure to deal with the associated commutation error