Lattice Boltzmann simulation of rarefied gas flows in microchannels

Abstract

For gas flows in microchannels, slip motion at the solid surface can occur even if the Mach number is negligibly small. Since the Knudsen number of the gas flow in a long microchannel can vary widely and the Navier-Stokes equations are not valid for Knudsen numbers beyond 0.1, an alternative method that can be applicable to continuum, slip and transition flow regimes is highly desirable. The lattice Boltzmann equation (LBE) approach has recently been expected to have such potential. However, some hurdles need to be overcome before it can be applied to simulate rarefied gas flows. The first major hurdle is to accurately model the gas molecule and wall surface interactions. In addition, the Knudsen number needs to be clearly defined in terms of LBE properties to ensure that the LBE simulation results can be checked against experimental measurements and other simulation results. In this paper, the Maxwellian scattering kernel is adopted to address the gas molecule and surface interactions with an accommodation coefficient (in addition to the Knudsen number) controlling the amount of slip motion. The Knudsen number is derived consistently with the macroscopic property based definition. The simulation results of the present LBE model are in quantitative agreement with the established theory in the slip flow regime. In the transition flow regime, the model captures the Knudsen minimum phenomenon qualitatively. Therefore, the LBE can be a competitive method for simulation of rarefied gas flows in microdevices

Capturing Knudsen layer phenomena using a lattice Boltzmann model

“…Examples include bounce-back, specular reflection, or a combination of the two [15,16,19,23,41], kinetic theory boundary conditions [17,42,43], and a virtual wall collision scheme [24]. In the present investigation, a kinetic boundary condition [17,42,44] has been used with the assumption of fully diffuse molecular reflection:…”

“…The method retains a computational efficiency comparable to Navier-Stokes solvers but is potentially a more accurate model for gas flows, over a broad range of Knudsen numbers, because its origins lie in kinetic theory. Since Nie et al [15] and Lim et al [16] first applied the lattice Boltzmann method to simulate rarefied gas flows, many publications have emerged which demonstrate that velocity slip and temperature jump phenomena can be captured by the lattice Boltzmann equation (LBE) approach [17][18][19][20][21][22][23][24][25][26]. However, the foregoing work focused on developing new boundary conditions for the velocity slip and temperature jump rather than constructing new LBE models that conserve symmetry for the higher-order moments (an essential requirement to obtain quantitative results for high Knudsen number flows).…”

Boundary induced nonlinearities at small Reynolds numbers

“…The subject of boundary condition for the Lattice Boltzmann Method has been extensively debated in the past years. An adequate set of kinetic boundary conditions has been proposed in [24] through a lattice transcription of the accommodation coefficients used in continuum kinetic theory of rarefied gases [35] and later similar kinetic boundary condition approaches have been proposed on more empirical grounds [23,36,37]. In particular in [22] it has been shown that the introduction of a slip function at the kinetic level is enough to produce, for small Knudsen numbers, local macroscopic boundary conditions of the form:…”

Lattice Boltzmann Simulations in the Slip and Transition Flow Regime with the Peano Framework

“…Virtual wall collisions of the Lattice Boltzmann particles are incorporated into the Lattice Boltzmann model, yielding satisfactory results for flow regimes up to Knudsen numbers ~30. Zhang et al report a successful qualitative Knudsen minimum prediction in [21]. Their results show good agreement for Knudsen numbers up to ~0.4 and only differ for higher numbers, due to numerical errors induced by the increasing value of the BGK-relaxationtime  .…”