The existing lattice Boltzmann method multispeed thermal models show a limited accuracy. This paper proposes a two-dimensional multispeed thermal model for the finite-difference lattice Boltzmann method (FDLBM). To recover correct fluid equations, up to fourth orders of local flow velocity should be retained in the local equilibrium distribution function and tensors of particle velocities should have up to seventh rank isotropy. In the FDLBM, particle velocities can be selected independently from the lattice configuration. Therefore, particle velocities of octagonal directions, which have up to seventh rank isotropic tensors, are adopted. The proposed model was verified by two simulations. The model showed excellent numerical stability in addition to strict accuracy.

No abstract

Multispeed thermal models of the lattice Boltzmann method (LBM) that have a single relaxation [Bhatnagar-Gross-Krook (BGK)] scheme have been proposed by several authors. While these models are intended to correctly represent heat characteristics and compressibility, most of them do not provide satisfactory accuracy. This paper discusses how to construct a correct model. Thermally correct two-dimensional and three-dimensional multispeed LBM BGK models are proposed. The models are verified by simulations of Couette flow, evolution from circularly distributed temperature, and normal shock wave. The results show exact agreement with the theoretical predictions. The numerical stability of the model is demonstrated by the simulation of recovery from a random fluctuation.

Two problems exist in the current studies on the application of the lattice Boltzmann method (LBM) to rarefied gas dynamics. First, most studies so far are applications of two-dimensional models. The numbers of velocity particles are small. Consequently, the boundary-condition methods of these studies are not directly applicable to a multispeed finite-difference lattice Boltzmann method (FDLBM) that has many velocity particles. Second, the LBM and FDLBM share their origins with the Boltzmann equation. Therefore, the results of LBM and FDLBM studies should be verified by the results of the continuous Boltzmann equation. In my review to date on the LBM studies, it appears that such verifications were seldom done. In this study, velocity slip and temperature jump simulations in the slip-flow regime were conducted using a three-dimensional FDLBM model. The results were compared with preceding theoretical studies based on the continuous Boltzmann equation. The results agreed with the theory with errors of a few percent. To further improve the accuracy of the FDLBM, it seems necessary to increase the number of velocity particles.

No abstract