Kinetic lattice Boltzmann method for microscale gas flows: Issues on boundary condition, relaxation time, and regularization

Abstract

It is well known that the Navier-Stokes equations cannot adequately describe gas flows in the transition and free-molecular regimes. In these regimes, the Boltzmann equation (BE) of kinetic theory is invoked to govern the flows. However, this equation cannot be solved easily, either by analytical techniques or by numerical methods. Hence, in order to efficiently maneuver around this equation for modeling microscale gas flows, a kinetic lattice Boltzmann method (LBM) has been introduced in recent years. This method is regarded as a numerical approach for solving the BE in discrete velocity space with Gauss-Hermite quadrature. In this paper, a systematic description of the kinetic LBM, including the lattice Boltzmann equation, the diffuse-scattering boundary condition for gas-surface interactions, and definition of the relaxation time, is provided. To capture the nonlinear effects due to the high-order moments and wall boundaries, an effective relaxation time and a modified regularization procedure of the nonequilibrium part of the distribution function are further presented based on previous work [Guo et al., J. Appl. Phys. 99, 074903 (2006); Shan et al., J. Fluid Mech. 550, 413 (2006)]. The capability of the kinetic LBM of simulating microscale gas flows is illustrated based on the numerical investigations of micro Couette and force-driven Poiseuille flows.

Lattice Boltzmann modeling of microchannel flows in the transition flow regime

“…9 are only valid in unbounded gas flow systems (Guo et al 2006). In a rarefied gas flow system confined by solid walls, some gas molecules collide with the walls and their flight paths will be shorter than the mean-free-path defined in unbounded systems (Niu et al 2007;Guo et al 2006Zhang et al 2006Zhang et al , 2007Tang et al 2008;Kim et al 2008;Stops 1970). As previously mentioned, some corrections on the mean-free-path and/or the viscosity have been developed to reflect the effect of gas molecule/wall interactions.…”

“…In recent efforts, the main focus is on developing LB models for rarefied gas flows with moderate and/or high Knudsen numbers. Since the LB equation is a discrete approximation to the continuous Boltzmann equation which can describe gas flows in an arbitrary flow regime, the usual approach is to design higher-order LB models via increasing the number of discrete velocities (Shan et al 2006;Ansumali et al 2007; Karlin and Ansumali 2007;Niu et al 2007). Nevertheless, Kim et al (2008) and Meng and Zhang (2009) recently found that the accuracy of higher-order LB models for rarefied gas flows does not monotonically increase with the order of the Guass-Hermite quadrature and more discrete velocities cannot guarantee an improved accuracy.…”

High-order regularization in lattice-Boltzmann equations

“…Furthemore, Zhang et al 17 , Niu et al 39 , and Montessori et al 23 investigated simulation of finite Kn channel flows, i.e. Couette and Poiseuille flows, both in 2D and 3D, using third-order LBEs together with the regularization: they projected also the rank 3 nonequilibrium moments onto the subspaces H q which, however, does not comply with the general principle established in Sec.…”

Lattice Boltzmann simulation of nonequilibrium effects in oscillatory gas flow

“…In addition, the lattice Boltzmann model does not suffer from the closure and boundary condition problems associated with higher-order continuum approaches such as Grad's method of moments [35]. More recently, higher-order LB models [36][37][38] and LB models incorporating the wall effect on the local mean free path [16,30,33,39] have captured the nonlinear behavior of the stress and heat flux in the Knudsen layer. However, to the best of the authors' knowledge, the lattice Boltzmann model has not been applied to oscillatory nonequilibrium flows.…”