Lattice Boltzmann Simulations of Magnetohydrodynamic Flows on a Rectangular Grid using a Central Moments Formulation

Simulations of magnetohydrodynamic (MHD) flows, especially in wall-bounded flows involving the resolution of Hartmann layers, are particularly effective with numerical techniques based on stretched grids. Lattice Boltzmann methods (LBMs) are highly parallelizable local algorithms based on collide-and-stream steps. Here, we extend a prior LBM formulation for MHD, which was constructed for square grids, to handle rectangular lattice grids. Our approach is based on augmenting the equilibria of a vector distribution function with terms involving the grid aspect ratio obtained via a Chapman-Enskog analysis in such a way that it consistently solves the magnetic induction equation. Similarly, the equilibria of another scalar distribution function representing the electrically conducting fluid motion subjected to the Lorentz force are extended with corrections terms based on the aspect ratio so that it correctly recovers its isotropy. For robust simulations, the collision steps of both the scalar and vector distribution functions are based on the relaxation of their central moments to the corresponding equilibria. Computations of various MHD flows at different Hartmann and magnetic Prandtl numbers based on our novel LB formulation demonstrates its accuracy and effectiveness.