Edge-Based Finite Element Formulation of Magnetohydrodynamics at High Mach Numbers

Abstract

In the context of the development of HALO3D (High Altitude Low Orbit 3D), an edge-based Finite Element Method multidisciplinary solver for hypersonic flows, this paper presents the simulation of flows with an imposed magnetic field. Applying the low-magnetic Reynolds number approximation, a current-continuity equation replaces Maxwell's equations with the Hall effect introduced through an electrical conductivity tensor. The approach is validated via the simulation of ionized flow through singly-paired segmented electrodes. Viscous, ionized flow simulations over a hemisphere at Mach 21.38 and over the OREX (Orbital Re-entry Experiment) capsule at Mach 17.61 are also presented. The 3D unstructured hybrid meshes used for these cases are optimized with a highly anisotropic methodology based on the Hessians of the solution, combining multi-physics adaptation criteria such as density, pressure, temperature, velocity, turbulent viscosity and electric potential. Unlike uniform mesh refinement or gradient-based mesh enrichment, this approach greatly improves accuracy without increasing mesh size.

KEYWORDS:

• MHD
• anisotropic mesh optimization
• edge-based finite element formulation

Acknowledgments

The authors would like to acknowledge the financial support of the NSERC-Lockheed Martin-Bell Helicopter Industrial Research Chair for Multi-physics Analysis and Design of Aerospace Systems, held at the McGill CFD Lab.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Natural Sciences and Engineering Research Council of Canada; Lockheed Martin Corporation..