High Resolution Schemes for Model Boltzmann Equation
Date Issued
2005
Date
2005
Author(s)
Tang, Gwo-Liang
DOI
zh-TW
Abstract
The Boltzmann equation is a nonlinear, integral, and differential equation with many variables. It is difficult to be solved mathematically, so the collision term is usually replaced with a collision model. This will make it easier to deal with. In this paper, the velocity space will be discreted by applying discrete ordinate method. The relation between velocity space and distribution function is eliminated, so the distribution function can be represented as proper discrete velocity points. Therefore, the motion equation of distribution function, which is continuous in physical space, velocity space, and time, is an integral and differential equation, and by discrete ordinate method it becomes differential equations, which are continuous in physical space and time only and point-wise in velocity space. After this kind of treatment, the difficulties of numerical calculating will be greatly reduced. In this paper, the WENO scheme in conjuction with discrete ordinate method was applied to solve the model Blotzmann equation, and the implicit WENO scheme for the model Blotzmann equation was developed to solve the steady solutions of rarefied gas flows.
First, the accuracy of the present scheme was verified by calculating the case of 1-D shock tube problem, which applied discrete ordinate method to discretize the velocity space of Blotzmann model equation and WENO scheme. The result of this case was also compared with results of other high resolution schemes.
Because it is difficult to describe the behaviors of collisions between different species of gas molecule, the collision frequency of different species of gas molecule was first developed and substituted into Blotzmann model equation to solve the binary gas mixture flow problem. The suitability was verified by comparing the result of 1-D shock tube case with the analytic solution of Euler’s equation in low Knudsen number condition. The collision frequency developed in this paper can surely describe the behaviors of gas molecules via the result.
In cases of 2-D flow problems, the external flows of cylinder and NACA 0012 airfoil were studied. For gas flow past cylinder, the characters of flow field in different Mach number and Knudsen number condition were investigated, and especially for low Knudsen number cases, the results were compared with calculating results of Euler’s equation. It showed that they are correspondent by comparing the characters of bow shock and wake. The convergence rates of different high resolution and implicit schemes were also investigated. The convergence behavior of the implicit WENO scheme developed in this paper is better than others. For gas flow past NACA 0012 airfoil, the calculating results were compared with results of experiment. It showed that the results of WENO scheme are of higher accuracy for the case with angle of attack.
Subjects
波茲曼模型方程式
The Model Boltzmann Equation
High Resolution Scheme
Type
thesis
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