In this work, we present a positivity-preserving high-order flux reconstruction method for the polyatomic Boltzmann--BGK equation augmented with a discrete velocity model that ensures the scheme is discretely conservative. Through modeling the internal degrees of freedom, the approach is further extended to polyatomic molecules and can encompass arbitrary constitutive laws. The approach is validated on a series of large-scale complex numerical experiments, ranging from shock-dominated flows computed on unstructured grids to direct numerical simulation of three-dimensional compressible turbulent flows, the latter of which is the first instance of such a flow computed by directly solving the Boltzmann equation. The results show the ability of the scheme to directly resolve shock structures without any ad hoc numerical shock capturing method and correctly approximate turbulent flow phenomena in a consistent manner with the hydrodynamic equations.

翻译：本文针对多原子玻尔兹曼-BGK方程提出了一种正性保持的高阶通量重构方法，并结合离散速度模型确保格式具备离散守恒性。通过模拟内部自由度，该方法可进一步推广至多原子分子，并涵盖任意本构关系。该方法在一系列大规模复杂数值实验中得到验证，涵盖基于非结构网格的激波主导流动，以及三维可压缩湍流的直接数值模拟（后者是首次通过直接求解玻尔兹曼方程计算此类流动）。结果表明，该格式无需任何特殊数值激波捕捉方法即可直接解析激波结构，并能以与流体动力学方程一致的方式正确近似湍流现象。