A comparative study of bandpass-filter-based multi-scale methods for turbulence energy cascade

Two different bandpass filtering methods have been used in the literature to study turbulence energy cascade simultaneously in physical and scale spaces. One of them is the derivative-based filter by Leung et al. [J Fluid Mech. 2012;V710:453–481] and the other is the difference-based filter by Motoori and Goto [J Fluid Mech. 2021;V911:A27]. The differences in their definitions lead to a variability in the quantitative results across the literature. In this work, we compare the two filtering methods and show that the difference-based filter can be related to the derivative-based filter through Taylor's approximation. Within the truncation error, the deviation between the two filters depends on the ‘filter width’, ‘a’, which is an input to the difference-based filter. The apparent reason for the deviations is that for large ‘a’, the difference-based filter extracts scales in a wider bandwidth compared to the derivative-based filter, which corresponds to the limiting case of the difference-based filter as a→1. Thus, one must be careful while choosing a filter based on their objectives, and while comparing results based on different filter designs. It is hoped that the present work will help reconcile the quantitative variations across the relevant literature.