On the convergence of the simplified Bernoulli trial collision scheme in rarefied Fourier flow

Abstract

The objective of this work is to provide a detailed study on the convergence behavior of the Simplified Bernoulli Trials (SBT) collision scheme in the direct simulation Monte Carlo (DSMC) method. One-dimensional Fourier heat conduction problem of argon gas at the early slip regime is considered. The problem consists of rarefied gas confined between two infinite parallel plates with different temperature magnitudes. The investigations compare the SBT solution for the Sonine-polynomial coefficients with theoretical predictions of the Chapman-Enskog theory. Also, the convergence behavior of the wall heat flux and the DSMC-calculated bulk thermal conductivity (KDSMC) are studied. The numerical performance of the DSMC method is affected by the number of computational particles (simulators) per cell, time step, and cell size. The dependence of the SBT collision scheme on discretization errors has been examined and compared with the no time counter (NTC) collision algorithm. Our results show that SBT captures analytical solutions of the Sonine polynomials using a few particles per cell. Unlike the NTC scheme, the SBT algorithm is not so sensitive to the number of simulators per cell, and the effective parameter in the convergence is the cell size to time step ratio, Δx/Δt, which should be adjusted properly for any specific test case. With setting a constant Δx/Δt, the SBT algorithm accurately predicts the wall heat flux solution by decreasing the average number of particles per cell to one particle or even less.