Development of a consistent and conservative scheme on a staggered grid for liquid metal MHD flows

A consistent and conservative scheme has been extended and developed on a staggered grid system for liquid metal MHD flow at a low magnetic Reynolds number by solving electrical potential Poisson equation based on the Ohm's law and the charge conservation law. The consistent scheme is used to ensure the calculated current density conserves the charge, and the divergence formula of the Lorentz force is used to ensure the momentum conservation. Simulation of liquid metal flows in a three-dimensional straight channel is conducted and compared with the analytical solutions from Shercliff's and Hunt's. The numerical results are in good agreement with analytical solutions for the Hartmann numbers from 50 to 5000. A fully conservative scheme on a staggered grid, which can conserve mass, momentum and kinetic energy and charge, is then developed with the central-symmetrical scheme for the convective term and the pressure term and with the consistent and conservative scheme for the calculation of the current density and the Lorentz force. A fully conservative scheme can be a good tool for numerical analysis of MHD flow instability, large eddy simulation (LES) and direct numerical simulation (DNS) of MHD turbulence.