Discrete ellipsoidal statistical BGK model and Burnett equations

To simulate non-equilibrium compressible flows, a new discrete Boltzmann model, discrete Ellipsoidal Statistical(ES)-BGK model, is proposed. Compared with the original discrete BGK model, the discrete ES-BGK has a flexible Prandtl number. For the discrete ES-BGK model in Burnett level, two kinds of discrete velocity model are introduced; the relations between non-equilibrium quantities and the viscous stress and heat flux in Burnett level are established. The model is verified via four benchmark tests. In addition, a new idea is introduced to recover the actual distribution function through the macroscopic quantities and their space derivatives. The recovery scheme works not only for discrete Boltzmann simulation but also for hydrodynamic ones, for example, based on the Navier-Stokes, the Burnett equations, etc.

💡 Research Summary

The paper introduces a novel discrete Boltzmann model – the discrete Ellipsoidal‑Statistical (ES) BGK model – designed to overcome the limitations of the conventional discrete BGK scheme when applied to non‑equilibrium compressible flows. The classic BGK collision operator forces the Prandtl number to unity, which prevents simultaneous control of viscosity and thermal conductivity. By replacing the post‑collision Maxwellian with an ellipsoidal Gaussian whose anisotropy is directly linked to a prescribed Prandtl number, the ES‑BGK model endows the simulation with a flexible Prandtl number while preserving the simplicity of the BGK framework.

To reach the Burnett level, the authors construct two families of discrete velocity models (DVMs). In two dimensions they augment the standard D2Q9 lattice with additional velocity vectors, forming a D2Q13‑type set that can exactly recover fourth‑order moments. In three dimensions a D3Q33 lattice (an extension of D3Q27) is employed, providing sufficient symmetry to capture both fourth‑ and fifth‑order moments required by the Burnett expansion. These DVMs guarantee that the discrete equilibrium moments match the continuous ones up to the order needed for Burnett‑level accuracy.

A Chapman‑Enskog expansion carried out to second order yields explicit relations between the non‑equilibrium tensors (the stress deviation Π_ij and the heat‑flux deviation Q_i) and the macroscopic viscous stress σ_ij and heat flux q_i, together with their spatial gradients and quadratic products. Symbolically, \