Multiscale discrete Maxwell boundary condition for the discrete unified gas kinetic scheme for all Knudsen number flows

TL;DR

This work addresses boundary conditions for multiscale gas flows by examining the discrete Maxwell boundary condition (DMBC) used in DUGKS and identifying discrete effects that cause nonphysical slip and jump in the continuum limit. It introduces a multiscale DMBC that enforcesCollision-free reflected distributions at the wall, restoring no-slip and no-jump as the relaxation time becomes small, while preserving accuracy across the free-molecular to continuum spectrum. The authors provide theoretical analysis in the free-molecular and continuum limits and validate the approach with numerical tests on isothermal Couette flow, Fourier flow, and lid-driven cavity flow, showing improved accuracy and reduced mesh sensitivity in the continuum regime. The method extends DUGKS to all Kn regimes and can be adapted to other kinetic models and multiscale transport problems such as phonon or radiation transfer.

Abstract

In this paper, a multiscale boundary condition for the discrete unified gas kinetic scheme (DUGKS) is developed for gas flows in all flow regimes. Based on the discrete Maxwell boundary condition (DMBC), this study addresses the limitations of the original DMBC used in DUGKS. Specifically, it is found that the DMBC produces spurious velocity slip and temperature jump, which are proportional to the mesh size and the momentum accommodation coefficient. The proposed multiscale DMBC is implemented by ensuring that the reflected original distribution function excludes collision effects. Theoretical analyses and numerous numerical tests show that the multiscale DMBC can achieve exactly the non-slip and non-jump conditions in the continuum limit and accurately captures non-equilibrium phenomena across a wide range of Knudsen numbers. The results demonstrate that the DUGKS with the multiscale DMBC can work properly for wall boundary conditions in all flow regimes with a fixed discretization in both space and time, without limitations on the thickness of the Knudsen layer and relaxation time.

Figures (16)

• Figure 1: Schematic of two neighboring cells $V_j$ and $V_{j+1}$ centered at $\boldsymbol x_j$ and $\boldsymbol x_{j+1}$, respectively. $\boldsymbol x_b$ is the midpoint of the cell interface, and $\boldsymbol n$ is the outward norm unit vector of cell $V_{j}$ at $\boldsymbol x_b$.
• Figure 2: Schematic of typical velocity profile for the isothermal Couette flow with slip effect. The velocity exhibits a linear spatial profile in the bulk flow and a nonlinear profile near the wall in the Knudsen layer.
• Figure 3: Velocity profiles of the Couette flow at different Knudsen numbers $[k=(\sqrt{\pi}/2)\text{Kn}]$ obtained from different DMBCs (the fully diffuse-reflection condition, i.e., $\alpha = 1$) with $N_y = 10$ and $N_y = 100$ uniform cells. In this and the subsequent plots, "orig. DMBC" denotes results obtained using the DUGKS with the original DMBC, while "multi. DMBC" denotes results from the DUGKS employing the multiscale DMBC. The LBE data are from Ref. sone1990numerical.
• Figure 4: Velocity profiles of the Couette flow in the continuum flow regime ($k = 1.0\times 10^{-5}$) obtained from different DMBCs with $N_y = 10$ and $N_y = 100$ uniform cells for different accommodation coefficient: (a) $\alpha = 0.1$, (b) $\alpha = 0.5$, and (c) $\alpha = 1.0$.
• Figure 5: Normalized velocity slip profiles of Couette flow versus $k$ with $N_y = 10$ and $N_y = 100$ uniform cells for different accommodation coefficient: (a) $\alpha = 0.1$, (b) $\alpha = 0.5$, and (c) $\alpha = 1.0$.