Discrete ellipsoidal statistical BGK model and Burnett equations

📝 Abstract

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.

💡 Analysis

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.

📄 Content

Discrete ellipsoidal statistical BGK model and Burnett equations Yudong Zhang1,2, Aiguo Xu2,3*, Guangcai Zhang2, Zhihua Chen1 Pei Wang2 1 Key Laboratory of Transient Physics, Nanjing University of Science and Technology, Nanjing 210094, China 2 Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China 3 Center for Applied Physics and Technology, MOE Key Center for High Energy Density Physics Simulations, College of Engineering, Peking University, Beijing 100871, China *Corresponding author. E-mail: Xu_Aiguo@iapcm.ac.cn Abstract A new discrete Boltzmann model, discrete Ellipsoidal Statistical(ES)-BGK model, is proposed to simulate non-equilibrium compressible flows. 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.

Key words: discrete Boltzmann model, ellipsoidal statistical BGK, Burnett equations, non-equilibrium quantities, actual distribution function.

1. Introduction Rarefied gas flows are traditionally associated with spacecraft re-entry into planetary atmosphere where the air is so thin that the applicability of Navier-Stokes (NS) model is challenged[1-3]. Recently, the rarefied effect of flows in microchannels have attracted significant research interest due to the rapid development of micro-fluidic technologies such as Micro-Electro-Mechanical System(MEMS)[4,5]. Generally, the rarefaction of flows can be measured by a dimensionless parameter, the Knudsen number (Kn), which is defined as the ratio of the mean free path of molecules to characteristic length that we focus on. In summary, there are two types of rarefied gas flows, one is the thin gas which has a large molecular distance, such as the air in the high altitude atmosphere[1], and the other is the flows with small characteristic length, such as shock wave and MEMS. In fact, according to the value of Kn, the flow can be divided into four categories including continuum flow(Kn<0.001), slip flow (0.001<Kn<0.1), transitional flow(0.1<Kn<10), and free molecular flow (Kn>10)[1,4,5].
   As we know, NS equations are applicable to continuum and slip flow (with slip boundary conditions), but it fails to provide the correct viscous stress and heat flux in transitional regime. The reason for the inapplicability of NS equations in transitional flow is that the constitutive equations, i.e. Newton’s viscosity law and the Fourier heat conduction law, are assumed to be linear which is inapposite when the non-equilibrium (or rarefaction) effect is significant. The Burnett equations[3,6], which are obtained from the Boltzmann equation through Chapman-Enskog (CE) expansion, have a modified constitutive equations and can work in part of the transition flow zones. However, the Burnett equations often encounter numerical instabilities because of the high order derivatives in the viscosity and heat flux terms[6]. It has been known that Boltzmann equation is applicable for all of the four flow regimes mentioned above. Unfortunately, the original Boltzmann equation is too complicated to be solved directly[7]. The multidimensional nature of distribution function and collision operator pose a great challenge for its numerical solution. Realistic numerical computations of the Boltzmann equation are based on probabilistic methods, such as direct simulation Monte Carlo (DSMC) method[8], or deterministic fast numerical methods, such as fast spectral method (FSM)[7]. In general, however, the computation cost is still too expensive for the direct solution of the Boltzmann equation. So, a variety of simplified methods have been developed to approximate the solution of the Boltzmann equation, such as the unified gas-kinetic scheme(UGKS)[9,10], the discrete velocity method (DVM)[11], the discrete unified gas-kinetic scheme (DUGKS)[12,13], the lattice Boltzmann method (LBM)[14-18], and the discrete Boltzmann method(DBM)[19-21].
   Recently, DBM has been developed as a non-equilibrium flow simulation tool and has been widely used in various flow conditions including high speed compressible flow, multiphase flow[22], flow instability[23-24], combustion and detonation[25-26], etc. Besides the values and evolutions of conserved kinetic moments, the DBM presents also those of the most relevant non-conserved kinetic moments. The la

This content is AI-processed based on ArXiv data.