What does Finite Volume-based implicit filtering really resolve in Large-Eddy Simulations?

The study illustrated in this paper completes the topics initially investigated in Ref. [42], the aim being here to analyze the role of the integral-based Finite Volume (FV) discretizations in Large Eddy Simulations that exploit the implicit filtering approach. Specifically, a theoretical study on the effective shape and length of three-dimensional filters induced by some FV-based flux reconstructions is the object of this paper. For any integral-based flux reconstruction, one gets always an approximation of the top-hat filter kernel. This is not the case of the filters induced by the differential-based Finite Difference operators, such as those reported and analyzed in Refs. [16] and [24]. Considering the sub-filter resolution parameter Q = Δeff/h, being Δeff the effective filter width and h the computational grid size, allows us discerning the effective measure of the approximate built-in top-hat filter. The induced shape and width is analyzed by means of a modified wavenumber-like analysis that is developed in the 3D Fourier space. Several evaluation criteria applied on different schemes are considered and the differences in terms of either velocity or flux interpolations on staggered or non-staggered grids are analyzed. Conclusions are reported that, depending on the using of either the integral or the differential form of the filtered equations, the induced numerical filter is or is not a congruent approximation of the exact top-hat transfer function for some value Q. The need of a suitable estimation of the sub-filter parameter Q is assessed from several real LES computations, that make use of the new integral-based version of the eddy-viscosity dynamic modeling presented in Ref. [42]. In fact, it is shown that the test-filtering length has to be carefully chosen as a function of the FV-based induced filter.