Mathematics

Numerical simulations of coarse-grained turbulence rely on two fundamental components: the filtering operator, which defines the level of coarseness in the representation, and the statistical operator, which extracts meaningful quantities from the chaotic computational data. The statistical operators typically applied in postprocessing numerical databases for statistically steady turbulence are a mixture of physical averages in homogeneous spatial directions and in time. Alternative averaging operators may involve phase or ensemble averages over different simulations of the same flow. This talk will introduce straightforward and easy-to-implement metrics to evaluate the relative importance of these averages, employing a mixed averaging analysis of variance. The application of these homogeneity indices will be demonstrated through three examples of steady turbulent flows, each exhibiting homogeneity in at least two spatial directions. The novel metrics provide valuable insights into identifying the most effective averaging procedures for flow configurations with two or more homogeneous directions. By optimizing the averaging process, they can contribute to achieving better statistics for turbulent flow predictions or reducing computing time. The research presented in this talk was made possible thanks to the invaluable collaboration and support of Professor Markus Klein from the University of the Bundeswehr Munich, Professors Andrea Ferrero and Francesco Larocca from Politecnico di Torino, Professors Guglielmo Scovazzi and Massimo Germano from Duke University.