In this work, closure of the Boltzmann–Bhatnagar-Gross-Krook (Boltzmann-BGK) moment hierarchy is accomplished via projection of the distribution function $f$ onto a space ${\mathbb{H}}^N$ spanned by $N$-order Hermite polynomials. While successive order approximations retain an increasing number of leading-order moments of $f$, the presented procedure produces a hierarchy of (single) $N$-order partial-differential equations providing exact analytical description of the hydrodynamics rendered by ($N$-order) lattice Boltzmann-BGK (LBBGK) simulation. Numerical analysis is performed with LBBGK models and direct simulation Monte Carlo for the case of a sinusoidal shear wave (Kolmogorov flow) in a wide range of Weissenberg number $\text{Wi}=\tau \nu k^2$ (i.e., Knudsen number $\text{Kn}=\lambda k=\sqrt{\text{Wi}}$); $k$ is the wave number, $\tau$ is the relaxation time of the system, and $\lambda \simeq \tau c_s$ is the mean-free path, where $c_s$ is the speed of sound. The present results elucidate the applicability of LBBGK simulation under general nonequilibrium conditions.

• Received 6 September 2009
• Corrected 12 February 2010

DOI:https://doi.org/10.1103/PhysRevE.81.026702