Abstract
The flow regime of micro flow varies from collisionless regime to hydrodynamic regime according to the Knudsen number Kn, which is defined as the ratio of the mean free path over the local characteristic length. On the kinetic scale, the dynamics of a small-perturbed micro flow can be described by the linearized kinetic equation. In the continuum regime, according to the Chapman-Enskog theory, hydrodynamic equations such as linearized Euler equations and Navier-Stokes equations can be derived from the linearized kinetic equation. In this paper, we are going to propose a unified gas kinetic scheme (UGKS) based on the linearized kinetic equation. For the simulation of small-perturbed micro flow, the linearized scheme is more efficient than the nonlinear one. In the continuum regime, the cell size and time step of UGKS are not restricted to be less than the particle mean free path and collision time, and the UGKS becomes much more efficient than the traditional upwind-flux-based operator-splitting kinetic solvers. The important methodology of UGKS is the following. Firstly, the evolution of microscopic distribution function is coupled with the evolution of macroscopic flow quantities. Secondly, the numerical flux of UGKS is constructed based on the integral solution of kinetic equation, which provides a genuinely multiscale and multidimensional numerical flux. The UGKS recovers the solution of linear kinetic equation in the rarefied regime, and converges to the solution of the linear hydrodynamic equations in the continuum regime. An outstanding feature of UGKS is its capability of capturing the accurate viscous solution in bulk flow region once the hydrodynamic flow structure can be resolved by the cell size even when the cell size is much larger than the kinetic length scale, such as the capturing of the viscous boundary layer with a cell size being much larger than the particle mean free path. Such a multiscale property is called unified preserving (UP) which has been studied in (Guo, et al. arXiv preprint arXiv:1909.04923, 2019). In this paper, a mathematical proof of the UP property for UGKS will be presented and this property is applicable to UGKS for solving both linear and nonlinear kinetic equations.
Highlights
1 Introduction The Boltzmann equation is a fundamental equation in kinetic theory, and it is widely applied in the fields of aerospace engineering, chemical industry, as well as the microelectromechanical systems (MEMS)
The linearized unified gas kinetic scheme (UGKS) well captures the solution of linear kinetic equation
The viscous solution can be accurately captured by UGKS even with cell size and time step being much larger than the particle mean free path and collision time
Summary
The Boltzmann equation is a fundamental equation in kinetic theory, and it is widely applied in the fields of aerospace engineering, chemical industry, as well as the microelectromechanical systems (MEMS). Similar to the asymptotic theories, the linearized Boltzmann equation has been studied when dealing with the small perturbed flow field in MEMS and porous media. The discrete unified gas-kinetic scheme (DUGKS) developed by Guo, et al is a multiscale scheme [5, 20], and has been successfully applied in the field of micro flow [21, 22], gas mixture [23], gas-particle multiphase flow [24], phonon transport [25], radiation [26], etc. In order to measure the capability of numerical schemes in capturing the multiscale flow physics, Guo, et al proposes the concept of unified preserving (UP) property [1].
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