Simulation of micro flow in the transition regime using effective-viscosity-based multi-relaxation-time lattice Boltzmann model

Abstract

With the rapid development of micro-electro-mechanical systems (MEMS), microscale rarefied gas flows have received considerable attention in the past decades. Recently, the lattice Boltzmann method (LBM) emerges as a promising way to study the flow in MEMS for its kinetic nature and distinctive computational features. Various LBM models have been used to simulate the microscale and nanoscale flow, among which the two-dimensional and nine-velocities (D2Q9)-based LBM is most widely accepted due to its extremely simplicity and high efficiency. However, the D2Q9-based LBM encounters great difficulties in the transition regime due to the rarefaction effects on mean free path and gas viscosity. An effective way to improve the capability of the existing LBM model is to incorporate an effective viscosity into the relaxation time, which can improve the accuracy of LBM model while keeping the simplicity and efficiency of LBM. However, the existing D2Q9-based LBM models with effective viscosity cannot give satisfactory predictions of the none-equilibrium phenomenon at moderate or high Knudsen (Kn) number both in accuracy and efficiency. To solve the above problem, in this study, an effective mean free path function proposed by Dongari et al. (Dongari N, Zhang Y H, Reese J M 2011 J. Fluids Eng. 133 071101) via modular dynamics mean is introduced into the D2Q9 multi-relaxation-time lattice Boltzmann model (MRT-LBM) to account for the effect of Knudsen layer in transition flow regime, and the viscosity in the MRT-LBM model is modified correspondingly. The combination of the bounce-back and specular reflection boundary condition is used to deal with the velocity slip, and the relaxation time and the reflection coefficient are properly set to eliminate the numerical artifact on the boundaries as the kinetic boundary condition is used. Micro Couette flow at Kn=0.1-6.77, and periodic Poiseuille flow at Kn=0.1128-2.2568, respectively, are numerically investigated by using the proposed MRT-LBM model, and the numerical results, including the non-dimensional velocity profile and the mass flow rate, are verified by the direct simulation Monte~Carlo (DSMC) data, the linearized Boltzmann solutions and the existing LBM model. The calculation results demonstrate that in transition regime, with the increase of Knudsen number, the dimensionless slip velocity at the wall significantly increases. It is shown that the velocity profiles predicted by the present MRT-LBM model agree well with the DSMC data and linearized Boltzmann solutions up to Kn=4.5 in Couette flow, which is much more accurate than that obtained from the existing LBM model. And the present LBM model gives at least the same order of accuracy in the prediction of velocity profile and mass flow rate as the existing LBM model in periodic Poiseuille flow. What is more, the Knudsen minimum phenomenon of flow in the microchannel is successfully captured at around Kn=1. The results demonstrate that the proposed model can enhance the ability of LBM in capturing the non-equilibrium phenomenon in micro flow in the transition regime both in accuracy and efficiency.