Lattice kinetic simulations in three-dimensional magnetohydrodynamics

A lattice kinetic algorithm to simulate three-dimensional (3D) incompressible magnetohydrodynamics is presented. The fluid is monitored by a distribution function, which obeys a scalar kinetic equation, subject to an external force due to the imposed magnetic field. Following the work of Dellar [J. Comput. Phys. 179, 95 (2002)], the magnetic field is represented by a different three-component vector distribution function, which obeys a corresponding vector kinetic equation. Discretization of the 3D phase space is based on a 19-bit scheme for the hydrodynamic part and on a 7-bit scheme for the magnetic part. Numerical results for magnetohydrodynamic (MHD) flow in a rectangular duct with insulating and conducting walls provide excellent agreement with corresponding analytical solutions. The scheme maintains in all cases tested the MHD constraint $\mathbf{\nabla }·\mathbf{B}=0$ within machine round-off error.

Figure 1

Discrete velocity lattice for the (a) hydrodynamic (3DQ19) and (b) magnetic (3DM7) fields.

Figure 2

Configuration of duct flow under an initial magnetic field $\mathit{B}=(0,B_y,0)$.

Figure 3

Velocity profiles and induced magnetic field for various Ha numbers for the case of insulating side and Hartmann walls. Comparison between analytical (solid line) and computational $(\times )$ results.

Figure 4

Velocity profiles and induced magnetic field for various Ha numbers for the case of perfectly conducting Hartmann walls and insulating side walls. Comparison between analytical (solid line) and computational $(\times )$ results.