Higher Order Thermal Lattice Boltzmann Model

Lattice Boltzmann method (LBM) modelling of thermal flows, compressible and micro flows requires an accurate velocity space discretization. The sub optimality of Gauss-Hermite quadrature in this regard is well known [1]. Most of the thermal LBM in the past have suffered from instability due to lack of proper H-theorem and accuracy [2]. Motivated from these issues, the present work develops along the two works [3] and [4] and imposes an eighth higher order moment to get correct thermal physics. We show that this can be done by adding just 6 more velocities to D3Q27 model and obtain a ``multi-speed on lattice thermal LBM'' with 33 velocities in 3D and ${\cal{O}}(u^4)$ and ${\cal{O}}(T^4)$ accurate $f_{i}^{\rm eq}$ with a consistent H-theorem and inherent numerical stability. Simulations for Rayleigh-Bernard as well as velocity and temperature slip in micro flows matches with analytical results. Lid driven cavity set up for grid convergence is studied. Finally, a novel data structure is developed for HPC.\\[4pt] [1] X. Shan and X. He, Phys. Rev. Lett. 80, 65 (1998).\\[0pt] [2] G. McNamara, A. Garcia, and B. Alder, J. Stat. Phys. 81, 395 (1995).\\[0pt] [3] S. Chikatamarla and I. Karlin, Phys. Rev. E 79, 046701 (2009).\\[0pt] [4] W. Yudistiawan et al. Phys. Rev. E 82, 046701 (2010)