Comparison of different Gaussian quadrature rules for lattice Boltzmann simulations of noncontinuum Couette flows: From the slip to free molecular flow regimes

Abstract

The lattice Boltzmann (LB) method can be formulated directly from the Boltzmann equation with the Bhatnagar–Gross–Krook assumption. This kinetic origin stimulates wide interest in applying it to simulate flow problems beyond the continuum limit. In this article, such a thought is examined by simulating Couette flows from the slip to free molecular flow regimes using the LB models equipped with different discrete velocity spaces, derived from the half-range Gauss Hermite (HGH), Gauss Legendre (GL), Gauss Kronrod (GK), and Gauss Chebyshev first and second quadrature rules. It is found that the conventional HGH-based LB models well describe noncontinuum Couette flows in the slip and weak transition flow regimes. Nonetheless, they suffer from significant errors with the further increasing Knudsen number, even if a large number of discrete velocities have been employed. Their results contrast with those by the LB models derived from the other Gaussian quadrature rules, which have far better accuracy at large Knudsen numbers. In particular, the GL- and GK-based LB models well capture the velocity fields of Couette flows in the strong transition and free molecular flow regimes. These numerical simulations in this article highlight the importance of velocity discretization for the LB simulations at different Knudsen numbers. They reveal that the LB models based on the Gauss Hermite (GH) quadrature rule are not always the best choice for simulating low-speed bounded flows at moderate and large Knudsen numbers; under strong noncontinuum conditions, those non-GH-based LB models proposed in this article have yielded more accurate results.