Lattice Boltzmann Boundary Conditions for Flow, Convection-Diffusion and MHD Simulations

TL;DR

This paper introduces a unified derivation of boundary conditions for lattice Boltzmann simulations across multiple physical problems, enabling accurate modeling of complex boundary interactions in flow, convection-diffusion, and MHD systems.

Contribution

A general boundary scheme compatible with various physical processes and boundary layouts is proposed, improving accuracy and flexibility in lattice Boltzmann simulations.

Findings

Boundary schemes accurately match analytical solutions.

Compatible with arbitrary boundary-to-grid distances.

Effective in simulating moving and static boundaries.

Abstract

A general derivation is proposed for several boundary conditions arisen in the lattice Boltzmann simulations of various physical problems. Pair-wise moment conservations are proposed to enforce the boundary conditions with given macroscopic quantities, including the velocity and pressure in flow simulations, concentration in convection-diffusion (CD) simulations, as well as magnetic field components in magnetohydrodynamical (MHD) simulations. Additionally, the CD and MHD simulations might involve the Robin boundary condition for surface reactions and a Robin-like boundary condition for thin walls with finite electrical conductivities, respectively, both of which can be written in a form with a variable flux term. In this case, the proposed boundary scheme takes the flux term as an increment to the bounced distribution function and a reference frame transformation is used to obtain a correction term for moving boundaries. Spatial interpolation and extrapolation are used for arbitrary boundary locations between computational grid points. Due to using the same approach in derivations, the obtained boundary schemes for different physical processes in a coupled simulation are compatible for arbitrary boundary-to-grid distances (not limited to the popular half-grid boundary layout) and arbitrary moving speeds. Simulations using half-grid and full-grid boundary layouts for flat boundaries are conducted for demonstrations and validations. Moving boundaries are simulated in hydrodynamic and MHD flows, while static boundaries are used in the CD simulations with surface reactions. The numerical and analytical solutions are in excellent agreement in the studied cases. The proposed boundary schemes are also applied in simulating fully coupled MHD pipe flows of a curved boundary with various boundary-to-grid distances and excellent agreement with analytical solutions is also obtained.