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

A simple procedure to approximate the noncommutation terms that arise whenever it is necessary to use a variable scale filtering of the motion equations and to compensate directly the flow solutions from the commutation error is here presented. Such a situation usually concerns large eddy simulation of nonhomogeneous turbulent flows. The noncommutation of the average and differentiation operations leads to nonhomogeneous terms in the motion equations, that act as source terms of intensity which depend on the gradient of the filter scale and which, if neglected, induce a systematic error throughout the solution. Here the different noncommutation terms of the motion equation are determined as functions of the δ-gradient and of the δ-derivatives of the filtered variables. It is shown here that approximated noncommutation terms of the fourth order of accuracy, with respect to the filtering scale, can be obtained using series expansions in the filter width of approximations based on finite differences and introducing successive levels of filtering, which makes it suitable to use in conjunction with dynamic or mixed subgrid models. The procedure operates in a way which is independent of the type of filter in use and without increasing the differential order of the equations, which, on the contrary, would require additional boundary conditions. It is not necessary to introduce a mapping function of the nonuniform grid in the physical domain into a uniform grid in an infinite domain. A priori}tests on the turbulent channel flow (Re_τ=180 and 590) highlight the approximation capability of the present procedure. A numerical example is given, which draws attention to the nonlocal effects on the solution due to the lack of noncommutation terms in the motion equation and to the efficiency of the present procedure in reducing the commutation error on the solution.