Modified algorithms for curved and virtual boundaries in Lattice Boltzmann method applications based on tree grid

Boundary treatment algorithms [5,10–12], for the curved boundaries [5,10,11] and tree grid virtual ones [12] employed in lattice Boltzmann method (LBM) related applications, are very important to numerical performance. In order to make them [5,10–12] theoretically more accurate and related to the physical realities, two modified new algorithms are presented in this paper. For the curved wall boundary treatment, the novelty resides in the method to construct the distribution functions. Compared with other popular algorithms, a pre-processing procedure is introduced in the improved algorithm involving both linear inner and outer interpolations which covers more neighbouring points to collect comprehensive information. Based on the constructed distribution functions, the velocity components and density on the target points are calculated. Finally, using the velocities and density determined to calculate the equilibrium distribution functions, the non-equilibrium distribution functions are obtained through a combination of both inner and outer interpolation of the neighbouring related points. The embedded depth Δ of each wall boundary point is treated individually, not by a rough classification like, Δ<0.5||Δ≥0.5 or Δ<0.75||Δ≥0.75 used in some other algorithms. For the tree grid virtual boundary treatment, compared with a previous algorithm [12] introduced by present authors, the modified new algorithm considers comprehensive information from more neighbouring related points to interpolate velocities and density. The distribution functions are calculated through a bidirectional extrapolation both from coarse to fine mesh and from fine to coarse one. It is proved that these two improved algorithms are trustable and have a good numerical performance.