Lattice Boltzmann model approximated with finite difference expressions. (20th September 2017) - Dubois, François; Lallemand, Pierre; Obrecht, Christian; Mahdi Tekitek, Mohamed
Highlights: In the introduction, we recall well known facts about the lattice Boltzmann method and the proposed adaptation of the artificial compressibility method proposed by Asinari et al. In a first section, we define the models: D2Q9 lattice Boltzmann scheme and linkwise artificial compressibility method. The modal analysis shows that the ACM scheme has a fundamental problem with the sound velocity; there is a non-physical link between the viscosity v and the sound velocity c s : c s = 2 v . In a second section, we propose to use the basic idea of the ACM scheme but with improved formulae natural in the framework of the Taylor expansion method proposed by one of us. We evaluate numerically the θ parameters with finite differences. We propose three stencils, using respectively 3, 5 or 9 vertices. In the third section, we test the above methods for a shear wave, the Stokes modes in a disk and a Poiseuille flow. The five-point stencil gives reasonable result but the remaining viscosity is smaller with the original D2Q9 lattice Boltzmann scheme. Abstract: We show that the asymptotic properties of the link-wise artificial compressibility method are not compatible with a correct approximation of fluid properties. We propose to adapt the previous method through a framework suggested by the Taylor expansion method and to replace first order terms in the expansion by appropriate three or five points finite differences and to add non linear terms. The "FD-LBM" scheme obtained by this method is tested in two dimensions for shear wave, Stokes modes and Poiseuille flow. The results are compared with the usual lattice Boltzmann method in the framework of multiple relaxation times.