Logistic Map and Bifurcation Parameters of a New 3-D Kinetic-based Discrete Dynamical System for Sub-grid-scale Modeling*

Discrete dynamical systems (DDS) are of significant interest for their ability to model complex, turbulent-like behaviors while maintaining mathematical simplicity. Hydrodynamic-based DDSs have been employed for turbulence modeling. This form of turbulence model has no closure problem and its bifurcation parameters appearing on the sub-grid scales (SGS) have physical interpretations, and their values can be deduced by high-pass filtering of resolved-scale results. In this work, we develop a 3-D kinetic-based DDS using the lattice Boltzmann method, leveraging its capacities to handle complex flow domains and its suitability for GPU parallel computation. Such capacities are critical for solving intricate flow systems such as porous-media flows and cardiovascular blood flows in which pulsation presents. We derive a logistic-like map of the DDS from the lattice Boltzmann equation using the Galerkin procedure. The DDS features two bifurcation parameters: a splitting factor β, which separates the large and small scales, and a rescaling factor θ. We optimize the combination of β and θ using Lagrangian DNS data from five pipe flows, utilizing L¹ and L² norms, skewness, and flatness, with Taylor Microscale Reynolds Numbers ranging from 450 to 750. The parameter set is then verified using an independent DNS case. These results from this study suggest that the DDS has potential applicability in predicting SGS physics relevant to complex flows such as pulsatile flows and turbulence.