## Numerical solution of MHD mean-flow equations

Assuming we want to simulate MHD flows with an existing general flow solver (e.g. Fluent, CFX or Star-CD), we need to add a minimum of electromagnetic equations. If the externally applied magnetic field is alternating (AC), one would probably choose to implement the induction equation for the three components of the induced magnetic field. If the magnetic field is static (DC), however, we can get away with solving only for scalar electrostatic potential. In the following I will limit myself to the DC case.

In steel casting, the fluid flow can be considered to be incompressible and Newtonian. The so-called magnetic Reynolds number is usually well below unity, so that the distortion of the magnetic field caused by the fluid flow remains small compared with the applied external field, . Ohm's law then gives the electrical current as

 , (1)

whereis the electric potential, is the electrical conductivity and  is the velocity. We can assume that electrical charge is conserved, so that . Together with (1) this gives a Poisson equation for the electric potential,

 . (2)

Once the electric potential is known, the electric current is given by (1) and the Lorentz force acting on the fluid can be computed as

 . (3)

In most general flow solvers the Poisson-type equation (2) for electric potential can be implemented as a generic transport equation, with transient and convection terms switched off.

In a staggered grid arrangement a correct, current-conserving discretisation of the equation would usually follow naturally, with the electric potential given in the scalar nodes, while potential gradient and -terms are computed on the vector nodes on control-volume faces.

An accurate discretisation is less obvious in a co-located variable arrangement. To avoid large errors, the -term should be calculated with interpolated velocities on the control-volume faces.  Cell-centre values of both current and Lorentz force are then computed as averages of the facial values. The discretisation in a co-located control-volume arrangement is discussed at length in the technical report cited below.