A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part I: On a rectangular collocated grid system

A consistent, conservative and accurate scheme has been designed to calculate the current density and the Lorentz force by solving the electrical potential equation for magnetohydrodynamics (MHD) at low magnetic Reynolds numbers and high Hartmann numbers on a finite-volume structured collocated grid. In this collocated grid, velocity (u), pressure (p), and electrical potential (φ) are located in the grid center, while current fluxes are located on the cell faces. The calculation of current fluxes on the cell faces is conducted using a conservative scheme, which is consistent with the discretization scheme for the solution of electrical potential Poisson equation. A conservative interpolation is used to get the current density at the cell center, which is used to conduct the calculation of Lorentz force at the cell center for momentum equations. We will show that both “conservative” and “consistent” are important properties of the scheme to get an accurate result for high Hartmann number MHD flows with a strongly non-uniform mesh employed to resolve the Hartmann layers and side layers of Hunt’s conductive walls and Shercliff’s insulated walls. A general second-order projection method has been developed for the incompressible Navier–Stokes equations with the Lorentz force included. This projection method can accurately balance the pressure term and the Lorentz force for a fully developed core flow. This method can also simplify the pressure boundary conditions for MHD flows.