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

TL;DR

The paper tackles the challenge of designing boundary conditions for lattice Boltzmann simulations that are accurate, physically consistent, and compatible across multiple physics. It introduces a unified derivation based on pair-wise moment conservations, interpolation/extrapolation, flux increments, and moving-boundary corrections to handle velocity, pressure, concentration, and magnetic-field boundaries on arbitrary boundary-to-grid distances. The resulting schemes cover Robin-like and Shercliff-boundary conditions, accommodate moving and curved boundaries, and are shown to be highly accurate across a suite of test problems including channel flow, Stokes’ problem, diffusion with surface reactions, Hartmann–Couette flow, and fully coupled curved MHD pipe flows. The approach facilitates fully coupled multiphysics simulations with consistent boundary treatment and broad applicability to complex geometries. Overall, these boundary schemes enhance the versatility and reliability of LBM in practical multi-physics contexts.

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.

Figures (16)

• Figure 1: Schematic of the original bounceback scheme Bouzidi2001. For the propagations in the $\vec{e}_1$ and $\vec{e}_3$ directions, B is the first solid grid point near the boundary marked by the dashed line and point X, A is the first fluid grid point near the boundary, and E is the next fluid grid point used for interpolation. The propagation starts from the interpolation point Y ending at A in panel (a) but starts from A ending at Y in panel (b), both of which have bounceback at X.
• Figure 2: Schematic of the proposed schemes using moments conservation at boundaries. Unlike Fig. \ref{['fig-schematic1']} based on the bounceback scheme at the point X, the current analysis considers the propagations from the points Y and B towards X over a same distance, namely $\overline{YX}=\overline{BX}=(1-d)\Delta x$ in both panels.
• Figure 3: Schematic of the proposed M scheme for bounceback using a point M with $\overline{MX}\equiv0.5\Delta x$ to impose the flux term. The distribution function starting from M is interpolated/extrapolated from E and A and increased by the flux term (and the correction term for moving boundaries, if any) when returning back to M. The unknown distribution function at A is then extrapolated/interpolated from E and M.
• Figure 4: Schematic of the 2D Hartmann--Couette flow arXiv2021.
• Figure 5: Three-dimensional channel flow shown in panel (a). Comparisons between the streamwise velocities $u_3$ obtained using the proposed boundary schemes (solid red) and the non-equilibrium extrapolation boundary scheme Guo2002 (dashed black) are presented in panel (b) using Eqs. \ref{['Eq-shift2-for velocity-final']} and \ref{['Eq-shift2-for pressure-final']} for a half-grid boundary layout with $d=0.5$ and panel (c) using Eqs. \ref{['Eq-shift1-BFL-for velocity-final']} and \ref{['Eq-shift1-BFL-for pressure-final']} for a full-grid boundary layout with $d=1$.