A positivity-preserving and conservative high-order flux reconstruction method for the polyatomic Boltzmann--BGK equation

TL;DR

This paper introduces a high-order flux reconstruction scheme for the polyatomic Boltzmann--BGK equation that preserves positivity and conserves mass, enabling accurate simulation of complex shock and turbulence phenomena.

Contribution

The method extends flux reconstruction to polyatomic molecules with arbitrary internal degrees of freedom, ensuring positivity and conservation in large-scale simulations.

Findings

Successfully simulates shock structures without ad hoc methods

First direct simulation of 3D compressible turbulent flows using Boltzmann equation

Accurately captures turbulence phenomena consistent with hydrodynamic equations

Abstract

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.