## Abstract

A novel numerical algorithm has been developed to solve the incompressible resistive magnetohydrodynamics equations in a fully coupled form. The numerical method is based on the face-centered unstructured finite volume approximation, where the velocity and magnetic field vector components are defined at the center of edges/faces; meanwhile, the pressure term is defined at element centroid. In order to enforce a divergence-free magnetic field, the gradient of a scalar Lagrange multiplier is introduced into the induction equation. A special attention will be given to satisfy the continuity equation and the Gauss' law for magnetism within each element and the summation of the equations can be exactly reduced to the domain boundary. The first modification to the original algorithm involves the evaluation of the convective fluxes over the two neighboring elements, where the discrete continuity equations are exactly satisfied. The second modification is based on the neglecting electric field term from the Lorentz force in two dimensions. The resulting large-scale algebraic linear equations are solved in a fully coupled manner using the one- and two-level restricted additive Schwarz preconditioners to avoid any time step restrictions forced by stability requirements. The spatial convergence of the algorithm is confirmed by solving the Hartmann flow, and then the algorithm is applied to the classical lid-driven cavity and backward facing step benchmark problems in two and three dimensions. The lid-driven cavity flow calculations at relatively high Stuart numbers indicate the perfect braking effect of the magnetic field in two dimensions.

Original language | English |
---|---|

Pages (from-to) | 347-371 |

Number of pages | 25 |

Journal | International Journal for Numerical Methods in Fluids |

Volume | 92 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1 May 2020 |

### Bibliographical note

Publisher Copyright:© 2019 John Wiley & Sons, Ltd.

### Funding

The authors are grateful for the use of the computing resources provided by the National Center for High Performance Computing of Turkey (UYBHM) under grant number 10752009 and the computing facilities at TUBITAK‐ULAKBIM, High Performance and Grid Computing Center.

Funders | Funder number |
---|---|

National Center for High Performance Computing of Turkey | |

UYBHM | 10752009 |

## Keywords

- backward facing step
- face-centered unstructured finite volume method
- fully coupled (monolithic) approach
- incompressible magnetohydrodynamics (MHD)
- lid-driven cavity