On the consistency of three-dimensional magnetohydrodynamical lattice Boltzmann models

Abstract

The lattice Boltzmann method (LBM) for magnetohydrodynamics (MHD) was first developed for two-dimensional (2D) flows. The magnetic component of the Lorentz force is obtained from the divergence of the Maxwell stress tensor, which is recovered by an appropriate modification to the LBM equilibrium distribution function for the solution of the velocity field. Although this method has subsequently been applied to three-dimensional (3D) flows, we show that several common modifications of the 3D equilibrium distribution function cannot simultaneously recover the Maxwell stress tensor and conserve mass exactly. Our study brings attention to this theoretical issue since such inconsistent models have been used in recent 3D LBM MHD simulations. Both requirements can be simultaneously satisfied by incorporating an additional model term to the 3D equilibrium distribution function. Three variants of Hunt's flow in a rectangular channel are considered to illustrate the undesirable consequences of using inconsistent models: 1) flow driven by a pressure difference, 2) flow driven by an external body force, and 3) inclusion of a streamwise gradient in magnetic pressure. While spurious errors were observed using inconsistent models due to the violation of mass conservation or inaccuracy in the Maxwell stress tensor, the consistent model always achieves the correct solutions in these cases and is recommended for 3D MHD simulations.