Frequency-domain general synthetic iterative scheme for efficient simulation of oscillatory rarefied gas flows

TL;DR

The paper develops a frequency-domain general synthetic iterative scheme (GSIS) to efficiently simulate oscillatory rarefied gas flows, addressing the slow convergence of conventional kinetic solvers in near-continuum regimes. By coupling a mesoscopic Boltzmann solver with macroscopic synthetic equations and reformulating stress/heat-flux closure with high-order terms, the method achieves fast convergence and asymptotic preservation, enabling accurate solutions on coarse meshes. The approach is validated on oscillatory shear between eccentric cylinders and squeeze-film damping for MEMS devices, showing up to three orders of magnitude speedup over conventional schemes while preserving the NSF limit in the appropriate regime. These results highlight the practical impact of GSIS for reliable, efficient kinetic simulations of MEMS-scale, time-periodic gas flows across a wide range of Knudsen and Strouhal numbers.

Abstract

Oscillatory rarefied gas flows are frequently encountered in MEMS, and their efficient numerical simulation remains a major challenge due to the time dependent nature of the problem and the high dimensionality of the Boltzmann kinetic equation. Here, we address this challenge by focusing on the periodic steady state and solving the resulting problem using the frequency domain general synthetic iterative scheme (GSIS). The key idea of GSIS is to simultaneously solve the mesoscopic kinetic equation and the macroscopic synthetic equation. The kinetic equation provides high-order constitutive relations, beyond those given by the Newton law of viscosity and the Fourier law of heat conduction, to the synthetic equation. In turn, the synthetic equation, which converges to the periodic steady state much faster than the kinetic equation, boosts the evolution of the kinetic equation toward the periodic steady state. As a result, super convergence is achieved, together with an asymptotic preserving property that allows the use of coarse spatial grids. The analytical Fourier stability analysis and the Chapman-Enskog expansion, together with challenging numerical simulations, are employed to demonstrate the fast convergence and asymptotic-preserving properties of GSIS, revealing that it can be three orders of magnitude faster than conventional kinetic schemes in near continuum flow regimes.

Figures (8)

• Figure 1: Error decay rate of CIS and GSIS variants as functions of the Strouhal number $S$ for different Knudsen numbers $Kn$. For brevity, the original GSIS is denoted by GSIS and the frequency-domain GSIS by GSIS-F. In (c) and (d), the CIS decay rate is very close to unity for $S<4$; therefore, it is omitted for clarity and only GSIS and GSIS-F are shown.
• Figure 2: Oscillatory shear-driven flow between two eccentric cylinders. Top: Geometry and structured mesh. The yellow dashed lines indicate the locations where velocity profiles are extracted. Two meshes with $N_{\text{cell}}\approx 6,000$ (Left) and $12,000$ (Right) are employed. Middle: Contours of $\Re(u_x)$ and $\Re(u_y)$ for $\delta_{rp}=10$ and $S=1$. Bottom: Contours of $\Re(u_x)$ and $\Re(u_y)$ for $\delta_{rp}=1000$ and $S=0.001$. The black dashed lines denote the reference solution used for comparison.
• Figure 3: The centerline streamwise velocity profiles at different iteration steps, when $\delta_{rp}=1000$ and $S=0.001$. The reference NSF solution is computed on the same mesh.
• Figure 4: The real part of the horizontal velocity at $x=0$. From left to right: $\delta_{rp}=1,10,100$. The top and bottom rows show the velocity profiles above and below the inner cylinder, respectively.
• Figure 5: Amplitude and phase of shear force $\gamma$ versus Strouhal number $S$ for $\delta_{rp}=1,10,100$.