Analysis of kinetic models and macroscopic continuum equations for rarefied gas dynamics.

摘要

The Boltzmann equation is the basic equation to describe rarefied gas flows. Some kinetic models with simple expressions for the collision term have been proposed to reduce the mathematical complexity of the Boltzmann equation. All macroscopic continuum equations can be derived from the Boltzmann equation or kinetic models through the Chapman-Enskog method, Grad's moment method, etc.; This thesis is divided into three parts. In the first part, existing kinetic models (BGK model, ES-BGK model, v(C)-BGK model, S model, and Liu model), and two newly proposed v(C)-ES-BGK type kinetic models are described and compared, based on properties that need to be satisfied for a kinetic model. In the new models a meaningful expression for the collision frequency is used, while the important properties for a kinetic model are retained at the same time.; In the second part of this work, the kinetic models (BGK, ES-BGK, v(C)-BGK, and two new kinetic models) are tested numerically for one-dimensional shock waves and one-dimensional Couette flow. The numerical scheme used here is based on Mieussens's discrete velocity model (DVM). Computational results from the kinetic models are compared to results obtained from the Direct Simulation Monte Carlo method (DSMC). It is found that for hard sphere molecules the results obtained from the two new kinetic models are very similar, and located in between the results from the ES-BGK and the v(C)-BGK models, while for Maxwell molecules the two new kinetic models are identical to the ES-BGK model. For one-dimensional shock waves, results from the new kinetic model II fit best with results from DSMC; while for one-dimensional Couette flow, the ES-BGK model is suggested.; Also in the second part of the work, a modified numerical scheme is developed from Mieussens's original DVM. The basic idea is to use a linearized expression of the reference distribution function, instead of its exact expression, in the numerical scheme. Results from the modified scheme are very similar to the results from the original scheme for almost all done tests, while 20--40 percent of the computational time can be saved.; In the third part, several sets of macroscopic continuum equations are examined for one-dimensional steady state Couette flow. For not too large Knudsen numbers (Kn ≤ 0.1) in the transition regime, it is found that the original and slightly linearized regularized 13 moment equations give better results than Grad's original 13 moment equations, which, however, give better results than the Burnett equations, while the Navier-Stokes-Fourier equations give the worst results, which is in agreement with the expectation. For large Knudsen number situations (Kn > 0.1), it turns out that all macroscopic continuum equations tested fail in the accurate description of flows, while the Grad's 13 moment equations can still give better results than the Burnett equations.