Solving the Boltzmann equation deterministically by the fast spectral method: application to gas microflows

Abstract

Based on the fast spectral approximation to the Boltzmann collision operator, we present an accurate and efficient deterministic numerical method for solving the Boltzmann equation. First, the linearized Boltzmann equation is solved for Poiseuille and thermal creep flows, where the influence of different molecular models on the mass and heat flow rates is assessed, and the Onsager-Casimir relation at the microscopic level for large Knudsen numbers is demonstrated. Recent experimental measurements of mass flow rates along a rectangular tube with large aspect ratio are compared with numerical results for the linearized Boltzmann equation. Then, a number of two-dimensional microflows in the transition and free-molecular flow regimes are simulated using the nonlinear Boltzmann equation. The influence of the molecular model is discussed, as well as the applicability of the linearized Boltzmann equation. For thermally driven flows in the free-molecular regime, it is found that the magnitudes of the flow velocity are inversely proportional to the Knudsen number. The streamline patterns of thermal creep flow inside a closed rectangular channel are analysed in detail: when the Knudsen number is smaller than a critical value, the flow pattern can be predicted based on a linear superposition of the velocity profiles of linearized Poiseuille and thermal creep flows between parallel plates. For large Knudsen numbers, the flow pattern can be determined using the linearized Poiseuille and thermal creep velocity profiles at the critical Knudsen number. The critical Knudsen number is found to be related to the aspect ratio of the rectangular channel.

A kinetic model of the Boltzmann equation for non-vibrating polyatomic gases

“…Third, (4.3) is independent of the viscosity index ω, so the molecular model has no influence on the flow rates when using the Rykov model. When our model (3.1) is used, the linearized BCO in (4.4) shows that different molecular models have different flow rates, according to the results in Sharipov & Bertoldo (2009) and Wu et al (2014) for monatomic gases.…”

“…Usually we choose γ = 0, but to solve the linearized BCO we choose γ = (2ω − 1)/2 to double the computational efficiency (Wu et al 2013(Wu et al , 2014. Finally, the macroscopic quantities are calculated as: The BCO (3.2) can be solved by the fast spectral method (Wu et al 2013) with a computational cost of O(M 2 N 3 v log(N v )), while the other collision operators in (3.1) can be solved by the discrete velocity method (Huang & Giddens 1967) with a cost of O(N 3 v ).…”

“…and its fast spectral approximation can be found in Wu et al (2014). The macroscopic quantities of interest are U 1 = h 0 v 1 dv, q t1 = h 0 v 1 (v 2 − 5/2)dv, and q r1 = h 2 v 1 dv.…”

“…Also, they cannot capture the differences in flow properties for different molecular models even in isothermal flows. For instance, for monatomic gases, the Shakhov kinetic model (Shakhov 1968) predicts the same mass flow rate for linearized Poiseuille flow with various viscosity 26 L. Wu, C. White, T. J. Scanlon, J. M. Reese and Y. H. Zhang indices, while the Boltzmann equation shows different mass flow rates for different molecular models (viscosity indices) (Sharipov & Bertoldo 2009;Wu et al 2014).…”

“…However, its intricate collision operator makes a solution difficult to obtain by deterministic numerical methods. For monatomic gases, the computational cost of the Boltzmann collision operator (BCO) is usually of the order of N 7 v , although this can be reduced to O(M 2 N 3 v log N v ) using the fast spectral method for some special collision kernels (Mouhot & Pareschi 2006;Wu et al 2013;Wu, Reese & Zhang 2014), where M 2 and N v are the number of discrete solid angles and velocity grid points in each velocity direction, respectively.…”