Can we find steady-state solutions to multiscale rarefied gas flows within dozens of iterations?

Abstract

One of the central problems in the study of rarefied gas dynamics is to find the steady-state solution of the Boltzmann equation quickly. When the Knudsen number is large, i.e. the system is highly rarefied, the conventional iteration scheme can lead to convergence within a few iterations. However, when the Knudsen number is small, i.e. the flow falls in the nearcontinuum regime, hundreds of thousands iterations are needed, and yet the "converged" solutions are prone to be contaminated by accumulated error and large numerical dissipation. Recently, based on the gas kinetic models, the implicit unified gas kinetic scheme (UGKS) and its variants have significantly reduced the iterations in the near-continuum flow regime, but still much higher than that of the highly rarefied gas flows. In this paper, we put forward a general synthetic iteration scheme (GSIS) to find the steady-state solutions of general rarefied gas flows within dozens of iterations at any Knudsen number. The key ingredient of our scheme is that the macroscopic equations, which are solved together with the Boltzmann equation and help to adjust the velocity distribution function, not only asymptotically preserves the Navier-Stokes limit in the framework of Chapman-Enskog expansion, but also contain Newton's law for stress and Fourier's law for heat conduction explicitly. For this reason, like implicit UGKS, the constraint that the numerical cell size should be smaller than the mean free path of gas molecules is removed, but we do not need the complex evaluation of numerical flux at the cell interface. What's more, as the GSIS does not rely on the specific kinetic model/collision operator, it can be naturally extended to quickly find converged solutions for mixture flows and even flows involving chemical reactions. These two superior advantages are also expected to accelerate the slow convergence in simulation of near-continuum flows via the direct simulation Monte Carlo method and its low-variance version. * Wei Su and Lianhua Zhu contribute equally.

On the accuracy of macroscopic equations for linearized rarefied gas flows

“…Solid lines represent the accurate results from the Shakhov model (44) solved by the discrete velocity method [69], that is, doubling the number of spatial and discrete velocity points gives the same results. Solid squares (experimental data) are collected from Ref.…”

“…In the framework of Gauss-Hermite quadrature, adding more discrete velocity grids is not economic, as these grids will be distributed in the region |v 1 | > 3 where the VDF is zero! In the discrete velocity method, it has already shown that designing numerical quadrature that is more suitable for wall-bounded problems [2,48,68] will greatly increase the [69], while symbols are approximate solutions of the Shakhov model when the molecular velocity space v is discretized according to the Gauss-Hermite quadrature of order N accuracy while reduce the computational cost. This may hint that to derive more accurate moment equations with limited number of moments, more suitable basis functions rather than the Hermite polynomials should be used.…”

A Unified Gas-kinetic Scheme for Micro Flow Simulation Based on Linearized Kinetic Equation

“…We study the flow induced by a hot microbeam in the transitional flow regime by UGKS and compare with the solutions with the R26 moment method [33] and the GSIS [6]. The numerical setup is the same as Zhu et al [20].…”

“…For the last several decades, researchers has been trying to develop effective multiscale numerical schemes [4][5][6][7][8][9]. In 2010, Xu et al proposes the unified gas-kinetic scheme, which is the first genuine multiscale scheme being able to capture the viscous effect with cell size much larger than the kinetic scale.…”

“…is also a multiscale scheme [5,19], and has been successfully applied in the field of micro flow [20,21], gas mixture [22], gas-particle multiphase flow [23], phonon transport [24], radiation [25], etc. The general synthetic iteration scheme was first proposed by Wu et al for the steady state solution of the linearized kinetic eqaution [6], and is recently extended to the simulation of nonlinear kinetic equation and diatomic gas [26,27].…”