High-Order Hydrodynamics from Boltzmann-BGK

In this work, closure of the 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 (LBGK) simulation. Numerical analysis is performed with LBGK models and direct simulation Monte Carlo (DSMC) for the case of a sinusoidal shear wave (Kolmogorov flow) in a wide range of Weissenberg number $Wi=\tau \nu k^2$ (i.e. Knudsen number $Kn=\lambda k=\sqrt{Wi}$);$k$isthewavenumber,$\tau$therelaxationtimeofthesystem,$\lambda\simeq\tau c_s$themean-freepath,and$c_s$ the speed of sound. The present results elucidate the applicability of LBGK simulation under general non-equilibrium conditions.