Multi-Group Maximum Entropy Method: Modeling Translational Non-Equilibrium

Abstract

The most rigorous physical description of non-equilibrium gas dynamics is rooted in the numerical solution of the Boltzmann equation. Yet, the large number of degrees of freedom and the wide range of both spatial and temporal scales render these equations intractable for many relevant problems. This study constructs a reduced-order model for the Boltzmann equation, by combining coarse-graining modeling framework with the maximum entropy principle. This is accomplished by projecting the high-dimensional Boltzmann equation onto a carefully chosen lower-dimensional subspace, resulting from the discretization of the velocity space into sub-volumes. Within each sub-volume, the distribution function is reconstructed through the maximum entropy principle, ensuring compliance with the detailed balance. The resulting set of conservation equations comprises mass, momentum, and energy for each sub-volume, allowing for flexibility in the description of the velocity distribution function. This new set of governing equations, while retaining many of the mathematical characteristics of the conventional Navier-Stokes equations far outperforms them in terms of applicability. The proposed methodology is applied to the Bhatnagar, Gross, and Krook (BGK) formulation of the Boltzmann equation. To validate the model's accuracy, we simulate the non-equilibrium relaxation of a gas under spatially uniform conditions. Additionally, the model is used to analyze the shock structure of a 1-D standing shockwave across an extensive range of Mach numbers. Notably, both the non-equilibrium velocity distribution functions and macroscopic metrics derived from our model align remarkably with the direct solutions of the Boltzmann equation.