Half-range lattice Boltzmann models for the simulation of Couette flow using the Shakhov collision term

The three-dimensional Couette flow between parallel plates is addressed using mixed lattice Boltzmann models which implement the half-range and the full-range Gauss-Hermite quadratures on the Cartesian axes perpendicular and parallel to the walls, respectively. The ability of our models to simulate rarefied flows are validated through comparison against previously reported results obtained using the linearized Boltzmann-BGK equation for values of the Knudsen number (Kn) up to $100$. We find that recovering the non-linear part of the velocity profile (i.e., its deviation from a linear function) at ${\rm Kn} \gtrsim 1$ requires high quadrature orders. We then employ the Shakhov model for the collision term to obtain macroscopic profiles for Maxwell molecules using the standard $\mu \sim T^\omega$ law, as well as for monatomic Helium and Argon gases, modeled through ab-initio potentials, where the viscosity is recovered using the Sutherland model. We validate our implementation by comparison with DSMC results and find excellent match for all macroscopic quantities for ${\rm Kn} \lesssim 0.1$. At ${\rm Kn} \gtrsim 0.1$, small deviations can be seen in the profiles of the diagonal components of the pressure tensor, the heat flux parallel to the plates, and the velocity profile, as well as in the values of the velocity gradient at the channel center. We attribute these deviations to the limited applicability of the Shakhov collision model for highly out of equilibrium flows.