Numerical Simulations for Semi-Classical Boltzmann Equation Using Asymptotic Preserving Scheme
Date Issued
2015
Date
2015
Author(s)
Chuang, Min-Cheng
Abstract
This study is aimed at solving the semi-classical Boltzmann-BGK equation to figure out the characteristics of gas flow, especially for rarefied gases. The coupling transformation of both the unsteady one dimensional Sod shock tube and the unsteady two dimensional shock wave impinging upon a square cylinder were investigated numerically. In addition, in order to reduce the computational amount, an appropriated mechanism is applied in this study. To deal with the discontinuity existing in problem, the solution of the semi-classical Boltzmann-BGK equation, namely the velocity distribution function, was divided into two parts with the help of a smoothed dirac delta function. Modified semi-classical Boltzmann-BGK equations were derived and solved for them over the whole computational domain then; the sum of the two parts gives the velocity distribution function in the buffer region. Consequently no more interface conditions need considering and the simulation is largely simplified. Three types of the smoothing functions – linear, cosine, and hypertangent, were tested and the conversation effect in buffer zone were examined in this thesis. As far as numerical discretization is concerned, the discrete coordinate method is employed for the velocity space and a high resolution scheme, either Total Variation Diminishing (TVD) or Weighted Essentially Non Oscillatory (WENO), was utilized for the physical space. Finally the asymptotic preserving scheme is taken in this study as well, which makes the relaxation time independent of collision term of semi-classical Boltzmann-BGK equation, resulting in a significant reduction in the computational amount. Finally the flow fields of quantum gas described by Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein statistics were all simulated. From the test examples of one dimensional unsteady Sod shock wave tube and two dimensional unsteady shock wave impinging upon a square cylinder, the investigation shows a use of a smoothing function and a high resolution scheme combined with the asymptotic preserving scheme technique does help reducing the computational amount.
Subjects
Semi-Classical Boltzmann-BGK Equation
Smoothing Function
High Resolution Scheme
Asymptotic Preserving Scheme
Type
thesis
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