Reduced kinetic model of polyatomic gases using lattice boltzmann methods

Abstract

The Boltzmann collision kernel and the widely used Bhatnagar–Gross–Krook (BGK) model are limited to monatomic gases as they do not account for the internal molecular structure. However, many real gases such as nitrogen, oxygen and methane are polyatomic. Kinetic models of polyatomic gas typically account for the internal degrees of freedom at the level of the two particle distribution function. Close to the hydrodynamic limit, the internal (rotational) degrees of freedom tend to be well represented just by rotational kinetic energy density. We account for the rotational energy by augmenting the ellipsoidal statistical Bhatnagar–Gross–Krook (ES–BGK) model, an extension of the BGK model, at the level of the single-particle distribution function with an advection–diffusion–relaxation equation for the rotational energy. This reduced model respects the H theorem and recovers the compressible hydrodynamics for polyatomic gases as its macroscopic limit. As required for a polyatomic gas model, this extension of the ES–BGK model not only has the correct specific heat ratio but also allows for three independent tunable transport coefficients: thermal conductivity, shear viscosity and bulk viscosity. An energy-conserving lattice Boltzmann model based on a crystallographic lattice for the simulation of weakly compressible flows is also proposed. The theoretical requirements and the methodology to construct such a model are discussed. We demonstrate that the model recovers the isentropic sound speed in addition to the effects of viscous heating and heat flux dynamics.