Derivation of a 3-D Kinetic-Based Discrete Dynamic System and Assessment with DNS Data

We present a 3-D discrete dynamical system (DDS) representing particle distribution functions for incompressible flows, derived from lattice Boltzmann equations in Fourier space. Through numerical experiments, we investigate various combinations of bifurcation parameters, leading to both laminar and turbulent flow behaviors. These flow behaviors are identified by analyzing the patterns of power spectral density (PSD) in the time series data. In our exploration, we delve into the underlying physics governing the observed flow behaviors, including interactions among motion scales and energy transport from large to small scales. Furthermore, we derive the DDS using logistic maps and evaluate its ability to capture flow behavior, comparing it with direct numerical simulation (DNS) of steady pipe flows. The pipe flow covers a range of Reynolds numbers, from 1300 to 6300, encompassing laminar, transitional, and turbulent flows. A systematic analysis reveals the similarities in the PSDs of the time series data obtained from both DDS and DNS approaches, characterizing dynamic flow behavior. This analysis considers various statistical properties such as mean, maximum, minimum, norm, skewness, flatness, etc., pertaining to sub-grid scale flow motion. Based on our findings, we conclude that this DDS can be effectively incorporated into large eddy simulation (LES) models, providing vital dynamic sub-grid scale information for pulsatile flows. These results highlight the potential applicability of the DDS in enhancing LES for the study of complex flow phenomena.