Representative deflection angle for the single-deflection method of direct simulation Monte Carlo.

Abstract

The total cross section of binary collision is, in general, unbounded due to the long-range interations of molecules. It is conventional to truncate the small angle deflections of collisions. The present work suggests an alternative way of avoiding the difficulty of unboundedness. We employ the mean value theorem of definite integral over the deflection angle for the cross section. A series of numerical experiments were carried out to look for the representative collision cross section through which the single-angle simulation is amenable to the solution of the Boltzmann equation. Results show that the cross section should be 〈Σ〉=Σ_{D}^{2}/(2Σ_{D}-Σ_{μ}), and the representative deflection for the single-angle simulation be cos〈χ〉=Σ_{μ}/Σ_{D}-1, where Σ_{D} is the diffusion cross section and Σ_{μ} is the viscosity cross section. The single-angle computations for the inverse power law and the Lennard-Jones force law perfectly reproduce the conventional scattering algorithms for one-dimensional (1D) simulations of transport coefficients and 1D shock thickness. The computation costs for Lennard-Jones molecules are comparable to the costs for inverse power-law models.