Boundary scheme for linear heterogeneous surface reactions in the lattice Boltzmann method

Lattice Boltzmann equation for convection-diffusion flows with Neumann boundary condition

In this work, a lattice Boltzmann equation (LBE) is developed to convection-diffusion flows with Neumann boundary condition in complex geometry. The physical fluid domain together with the physical boundary is extended to a large domain similar to the fictitious domain method, and the original convection-diffusion equation (CDE) is reformulated for the large domain, where the Neumann boundary condition is naturally incorporated to CDE without separated explicit treatment. Based on this extended CDE for the large domain, the LBE solver is designed accordingly. Several classical simulations of pure thermal diffusion/natural convection between two concentrated cylinders, natural convection flow around a circular cylinder with constant heat flux in square cavity, and mixed convection flow in a square lid-driven cavity with a circular cylinder are carried out to validate the present method. Numerical results show that the predictions by present LBE agree well with theoretical or other results.

Mixed bounce-back boundary scheme of the general propagation lattice Boltzmann method for advection-diffusion equations

In this work, a mixed bounce-back boundary scheme of general propagation lattice Boltzmann (GPLB) model is proposed for isotropic advection-diffusion equations (ADEs) with Robin boundary condition, and a detailed asymptotic analysis is also conducted to show that the present boundary scheme for the straight walls has a second-order accuracy in space. In addition, several numerical examples, including the Helmholtz equation in a square domain, the diffusion equation with sinusoidal concentration gradient, one-dimensional transient ADE with Robin boundary and an ADE with a source term, are also considered. The results indicate that the numerical solutions agree well with the analytical ones, and the convergence rate is close to 2.0. Furthermore, through adjusting the two parameters in the GPLB model properly, the present boundary scheme can be more accurate than some existing lattice Boltzmann boundary schemes.

Consistent second-order boundary implementations for convection-diffusion lattice Boltzmann method

In this study, an alternative second-order boundary scheme is proposed under the framework of the convection-diffusion lattice Boltzmann (LB) method for both straight and curved geometries. With the proposed scheme, boundary implementations are developed for the Dirichlet, Neumann and linear Robin conditions in a consistent way. The Chapman-Enskog analysis and the Hermite polynomial expansion technique are first applied to derive the explicit expression for the general distribution function with second-order accuracy. Then, the macroscopic variables involved in the expression for the distribution function is determined by the prescribed macroscopic constraints and the known distribution functions after streaming [see the paragraph after Eq. (29) for the discussions of the "streaming step" in LB method]. After that, the unknown distribution functions are obtained from the derived macroscopic information at the boundary nodes. For straight boundaries, boundary nodes are directly placed at the physical boundary surface, and the present scheme is applied directly. When extending the present scheme to curved geometries, a local curvilinear coordinate system and first-order Taylor expansion are introduced to relate the macroscopic variables at the boundary nodes to the physical constraints at the curved boundary surface. In essence, the unknown distribution functions at the boundary node are derived from the known distribution functions at the same node in accordance with the macroscopic boundary conditions at the surface. Therefore, the advantages of the present boundary implementations are (i) the locality, i.e., no information from neighboring fluid nodes is required; (ii) the consistency, i.e., the physical boundary constraints are directly applied when determining the macroscopic variables at the boundary nodes, thus the three kinds of conditions are realized in a consistent way. It should be noted that the present focus is on two-dimensional cases, and theoretical derivations as well as the numerical validations are performed in the framework of the two-dimensional five-velocity lattice model.