Advection–Diffusion Lattice Boltzmann Method With and Without Dynamical Filter

Abstract

In multi-component flow and/or thermal flows, when the diffusion coefficient of the advection–diffusion equation is relatively small, the relaxation coefficient in the lattice Boltzmann method will be close to 0.5, which will lead to numerical instability. The stability conditions will become more severe, when there are high gradient regions in the computational domain. In order to improve the stability of advection–diffusion lattice Boltzmann method to simulate scalar transport in complex flow, a hybrid regularized collision operators and a dynamic filtering method which is suitable for the convection-diffusion lattice Boltzmann method are proposed in this paper. The advection–diffusion lattice Boltzmann method is first tested in uniform flow with smooth and discontinuous initial conditions. Then the scalar transport in doubly periodic shear layer flow is tested, which is sensitive to numerical stability. The adaptive dynamic filtering method is also tested. The results are compared to the classical finite difference method and to the lattice Boltzmann method using the projection-based regularized and standard Bahtnagar-Gross-Krook collision operator. The results show that the hybrid regularized collision operator has advantages in simulating the scalar advection-diffusion problem with small diffusion coefficient. In addition, the adaptive filtering method can also improve the numerical stability of the lattice Boltzmann method with limited numerical dissipation.