maelstrom

maelstrom is a numerical software tool for the solution of magnetohydrodynamics problems in cylindrical coordinates. As such, maelstrom includes time integrators for the heat equation, for the Navier--Stokes equations, and a stationary solver for the Maxwell equations, each in cylindrial coordinates.

Some details on the problem

The goal is to compute the flux of a liquid metal under the influence of a magnetic field, modeled by

• the heat equation,
• Maxwell's equations, and
• the Navier-Stokes equations.

Heat and Navier-Stokes are coupled by buoyancy, heat and Maxwell by the Joule effect, and Maxwell and Navier-Stokes by current induction and the Lorentz force.

To simplify matters, it is assumed that the effect of the material flux does not influence the electric and magnetic fields, i.e., the current induction from moving molten metal in a magnetic field is neglected. This decouples Maxwell's equations from the other two. Essentially, the task breaks down to

• computing Joule heating and Lorentz force, given a voltage distribution in coils, and given those two quantities
• computing the the resulting material flux inside a container.

Solving Maxwell's equations

Derivation of the involved formulas is best taken from the documentation.

Some visualizations

A typical cylindrical problem: A crucible with a liquid on the left, surrounded by a number of electric coils (the squares). The arrows indicate the magnetic field produced by current in those coils. Note that the actual domain where Maxwell's equations are solved is much larger.

The Joule heat source (blue/red) and the Lorentz force (arrows) generated from the above magnetic field.

The temperature in the first 60 seconds of the full simulation. The Maxwell equations are solved first, from this one gets the above Joule heat source and the Lorentz force. These are added as external fources to the Boussinesq simulation that we see here.

Testing

To run the voropy unit tests, check out this repository and type

pytest

License

maelstrom is published under the MIT license. See the file LICENSE for detailed information.