Castanon Quiroz, Daniel (2016-04). Solving the MHD Equations in the Presence of Non-Axisymmetric Conductors Using the Fourier-Finite Element Method. Doctoral Dissertation.
Thesis
The research presented in this dissertation focus on the numerical approximation of the magnetohydrodynamic (MHD) equations in the von K'arm'an Sodium (VKS) set-up. These studies are performed with the SFEMaNS MHD code developed by J.-L. Guermond and C. Nore since 2002 for axisymmetric geometries. SFEMaNS is based on a spectral decomposition in the azimuthal direction and a Lagrange finite element approximation in a meridian plane. To overcome the axisymmetric restrictions, we propose a novel numerical method to solve the Maxwell part of the MHD equations, and use a pseudo-penalty method to model the rotating impellers. We then present hydrodynamic and MHD simulations of the VKS set-up. Hydrodynamic results compare well with the experimental data in the same range of kinetic Reynolds numbers: at small impeller rotation frequency, the flow is steady; at larger frequency, the fluctuating flow is characterized by small scales and helical vortices localized between the blades. MHD computations are performed for two different flows. One with small kinetic Reynolds number, and the other with a larger one. In both cases, using a ferromagnetic material for the impellers decreases the dynamo threshold and enhances the predominantly axisymmetric magnetic field: the resulting dynamo is a mostly axisymmetric axial dipole with an azimuthal component concentrated in the impellers as observed in the VKS experiment.
