A fast spectral method for the Boltzmann equation for monatomic gas mixtures

Although the fast spectral method has been established for solving the Boltzmann equation for single-species monatomic gases, its extension to gas mixtures is not easy because of the non-unitary mass ratio between the di↵erent molecular species. The conventional spectral method can solve the Boltzmann collision operator for binary gas mixtures but with a computational cost of the order m3rN6, where mr is the mass ratio of the heavier to the lighter species, and N is the number of frequency nodes in each frequency direction. In this paper, we propose a fast spectral method for binary mixtures of monatomic gases that has a computational cost O(pmrM2N4 logN), where M2 is the number of discrete solid angles. The algorithm is validated by comparing numerical results with analytical Bobylev- Krook-Wu solutions for the spatially-homogeneous relaxation problem, for mr up to 36. In spatially-inhomogeneous problems, such as normal shock waves and planar Fourier/Couette flows, our results compare well with those of both the numerical kernel and the direct simulation Monte Carlo methods. As an application, a two-dimensional temperature-driven flow is investigated, for which other numerical methods find it difficult to resolve the flow field at large Knudsen numbers. The fast spectral method is accurate and elective in simulating highly rarefied gas flows, i.e. it captures the discontinuities and fine structures in the velocity distribution functions

Computing of gas flows in micro- and nanoscale channels on the base of the Boltzmann Kinetic equation

AbstractThe paper describes the methodology of computing gas flows in narrow micro- and nanoscale channels on the basis of finite-difference solution of the Boltzmann kinetic equation using the conservative projection method of collision integral calculation. Mathematical framework of the method is considered and the problem solving environment for calculation of the above mentioned flows is described. Examples of the flow calculations in the plane and 3D geometry are given