A new Magneto-Fluid Dynamics model for Low Magnetic Reynolds number regime in the Hybridized Discontinuous Galerkin Framework

The Magneto-Fluid Dynamics has many aerospace applications when the gas is an ionized mixture. Typically, this happens when the temperature of the mixture is very high. Indeed, we need to consider new degrees of freedom of the molecules, the presence of the chemical reactions, and the possibility to have free electrons that can move from one point to another. This is the typical situation during a normal atmospheric entry problem. In this context, all the physical properties cannot be computed with the classical approximation laws. The Chapman-Enskong method gives us the possibility to perturb the Boltzman Transport Equation and to understand which are the different contributions (of electrons and heavy particles) to take into account. Hence, the modeling of the transport properties can be conducted. The latter, which are functions of pressure and temperature, can also change their values with the presence of external magnetic and electric fields. The physical models that try to describe all these peculiar characteristics and the macroscopic properties of electrically conducting flows are, from a numerical point of view, highly stiff. In the last years, a simplified version of the full Magneto-Fluid Dynamics system was proposed. Two limiting cases are present based on the dimensionless parameter Magnetic Reynolds number. When it is very small, the externally applied magnetic field is called rigid. Hence, the induced magnetic field can be considered null. Therefore, the classical Navier-Stokes equations with additional external electromagnetic source terms make up the simplified Magneto-Fluid Dynamics system. The source term is composed of the Lorentz force for the momentum equation and the Joule heating for the energy equation. No source terms for the continuity equations must be taken into account under the assumption of frozen flow, which is generally adopted in many hypersonic applications. The system cannot be solved analytically, except where we are considering parallel flow, where the non-linear convective terms drop down, i.e. Hartmann flow. As a consequence, we need to introduce numerical discretization. One of the most innovative methods to solve non-linear PDEs is called Hybridizable Discontinuous Galerkin. It can be considered as a mix between FE and FV methods. The hybridization concept introduces the possibility to parallelize efficient the code and to reduce the high computational cost associated with the high-order methods, e.g. Discontinuous Galerkin. In the past years, many Magneto-Fluid Dynamics physical models were developed into FV solvers. From the literature review, only one researcher Group from the Massachusetts Institute of Technology and Imperial College London, in 2019, has developed an MHD model for compressible flow in their HDG solver. The High-Order Unifying Framework, developed by May et al. at RWTH Aachen and currently at The VKI, was empty of the Magneto-Fluid Dynamics model for low Magnetic Reynolds number. The implementation from a physical, mathematical, and computer science point of view, is explained in this work. Finally, the model has been tested with simple test cases. Comparison with analytical data, i.e. Hartmann flow, has been conducted to validate the model.