Magnetohydrodynamics (MHD)

Simcenter STAR-CCM+ allows you to model the interaction between electrically conducting fluids, such as molten metals, electrolytes, and plasmas, and electromagnetic fields.

A conducting fluid in relative motion to a magnetic field induces an electric current density:

where $\sigma$ is the electrical conductivity (see Eqn. (4228)), $v$ is the flow velocity, and $\mathbf{B}$ is the magnetic flux density. Eqn. (4369) contributes to the total electric current density $J$ (see Eqn. (4242)).

$J_L$ in turn induces a magnetic flux density which contributes to the total magnetic flux density in Eqn. (4369). Simcenter STAR-CCM+ calculates $J_L$ using either a one-way coupled or a two-way coupled approach. In the one-way coupled approach, Simcenter STAR-CCM+ calculates $J_L$ with respect to a prescribed magnetic flux density, ${\mathbf{B}}_p$ . In the two-way coupled approach, Simcenter STAR-CCM+ calculates the total magnetic flux density $B$ , which also accounts for the magnetic flux density induced by $J_L$ .

One-Way Coupled MHD

The ratio between the induced and the prescribed magnetic flux densities is defined by a non-dimensional number known as the magnetic Reynolds number:

$$R{e}_{mag}={\mu }_0\sigma UL$$

(4370)

where ${\mu }_0$ is the vacuum permeability, $U$ is the characteristic flow velocity and $L$ is the characteristic length.

The one-way coupled MHD approach is suitable for small $R{e}_{mag}$ , where the induced magnetic flux density is significantly weaker than the prescribed magnetic flux density.

In this case, Simcenter STAR-CCM+ solves Eqn. (4242) in a decoupled manner, neglecting the eddy currents induced by time-varying magnetic fields (see Low-Frequency Electromagnetics in Conducting Media). In absence of eddy currents, adding $J_L$ (Eqn. (4369)) to the total electric current density $J$ (Eqn. (4234)) gives:

$$J=-\sigma \nabla \varphi +\sigma v\times B$$

(4371)

where $B$ is the prescribed magnetic flux density:

$$B=B_p$$

(4372)

As $J=\sigma E$ (see Eqn. (4228)), the electric field is defined as:

$$E=-\nabla \varphi +v\times B$$

(4373)

Two-Way Coupled MHD

With this approach, Simcenter STAR-CCM+ solves Eqn. (4241) (and, when modeling the electric potential, Eqn. (4242)). In this case, the electric field is calculated as:

$$E=-\nabla \varphi -\frac{\partial A}{\partial t}+v\times \mathbf{B}$$

(4374)

The total electric current density is:

$$J=\sigma E=\sigma (-\nabla \varphi -\frac{\partial A}{\partial t}+v\times \mathbf{B})$$

(4375)

with:

$$B=\nabla \times A$$

(4376)

In quasi-unsteady analyses, the term $\partial A/\partial t$ in Eqn. (4374) and Eqn. (4375) is neglected. When Simcenter STAR-CCM+ does not solve for the electric potential, the term $\nabla \varphi$ in Eqn. (4374) and Eqn. (4375) is also neglected.

For simplicity, Eqn. (4371) and Eqn. (4375) assume that the only contribution to $J_{ex}$ in Eqn. (4234) is $J_L$ . Additional contributions to $J_{ex}$ are discussed in the relevant sections.

In circuit breaker simulations (plasmas), the contribution to the electric current density that comes from the gradient of the electric potential, $J_{\varphi }=-\sigma \nabla \varphi$ , typically dominates over the other contributions. In electromagnetic stirring applications, the dominant contribution typically comes from the eddy currents, $J_A=-\sigma \partial A/\partial t$ . When the magnetic flux density and the fluid velocity are significant, $J_L$ becomes the dominant term. This is typically the case in turbulence control applications, which use the Lorentz force to modulate or suppress the flow turbulence.