A 3-D Poor Man's Boltzmann Equation

A 1-D ``synthetic'' distribution function for the poor man’s Boltzmann equation (PMBE) has been studied previously; but real applications must be in three space dimensions. In this presentation we outline derivation of the 3-D PMBE and study bifurcations of the corresponding discrete dynamical system (DDS) for this case. In particular, we first provide a brief single-mode Galerkin derivation of the PMBE, and then present time series, power spectra and regime maps to demonstrate its consistency with expected fluid flow behaviors—in particular, existence of Ruelle \& Takens, Feigenbaum, and Pomeau \& Manneville bifurcation sequences, as well as combinations of these. We also suggest how such a DDS can be used to produce very efficient sub-grid scale synthetic distribution function models for turbulence simulations within a lattice-Boltzmann/large-eddy simulation framework.