Fokker–Planck model for computational studies of monatomic rarefied gas flows

Abstract

In this study, we propose a non-linear continuous stochastic velocity process for simulations of monatomic gas flows. The model equation is derived from a FokkerPlanck approximation of the Boltzmann equation. By introducing a cubic non-linear drift term, the model leads to the correct Prandtl number of 2/3 for monatomic gas, which is crucial to study heat transport phenomena. Moreover, a highly accurate scheme to evolve the computational particles in velocity-and physical space is devised. An important property of this integration scheme is that it ensures energy conservation and honours the tortuosity of particle trajectories. Especially in situations with small to moderate Knudsen numbers, this allows to proceed with much larger time steps than with direct simulation Monte Carlo (DSMC), i.e. the mean collision time not necessarily has to be resolved, and thus leads to more efficient simulations. Another computational advantage is that no direct collisions have to be calculated in the proposed algorithm. For validation, different micro-channel flow test cases in the near continuum and transitional regimes were considered. Detailed comparisons with DSMC for Knudsen numbers between 0.07 and 2 reveal that the new solution algorithm based on the Fokker-Planck approximation for the collision operator can accurately predict molecular stresses and heat flux and thus also gas velocity and temperature profiles. Moreover, for the Knudsen Paradox, it is shown that good agreement with DSMC is achieved up to a Knudsen number of about 5.

Analysis of the thermomechanical inconsistency of some extended hydrodynamic models at high Knudsen number

“…Gorji et al [32] reported very good agreement of their stochastic kinetic equation (44) with experiments for a slip/transition regime flow as did Bogomolov and Dorodnitsyn [10].…”

“…In [31,32], a kinetic equation is proposed that consists of replacing the Boltzmann collision integral with a velocity space stochastic operator. The proposed kinetic equation is written:…”

“…The distribution function f in equation (41) is primarily a function of the macroscopic time and spatial position variables, denoted t and x respectively (see page 4 of [32]). …”

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

“…In these models, the gain part of the BCO is modelled by the Gauss, ellipsoidal Gauss, and Gauss-Hermite polynomials, while the loss part describes the exponential decay of the distribution function with a rate independent of molecular velocity. Recently, Gorji and co-workers have also proposed a model replacing the BCO by the Fokker-Planck collision operator (Gorji, Torrilhon & Jenny 2011;Gorji & Jenny 2013), which models the drift and diffusion in velocity space. Although this model is faster than the DSMC method near the continuum-fluid regime, for microflow simulations it suffers the same slowness as the DSMC method because of its particulate nature.…”

Transition regime analytical solution to gas mass flow rate in a rectangular micro channel

“…Among recent developments in predicting these phenomena, Gallis and Torczynski [8] have presented a direct Monte Carlo simulation (DSMC). Gorji et al [9] obtained the mass flowrate by solving, numerically, a velocity-space stochastic equation addressing molecular motions. Veltzke and Thaming [10] pointed out that the mass flowrate in the slip flow regime can be accurately predicted by arguments of molecular spatial diffusion effects without using any fitting parameter.…”