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A face‐based monolithic approach for the incompressible magnetohydrodynamics equations
Author(s) -
Ata Kayhan,
Sahin Mehmet
Publication year - 2020
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.4786
Subject(s) - mathematics , discretization , finite element method , mathematical analysis , lorentz force , lagrange multiplier , multigrid method , magnetohydrodynamics , induction equation , maxwell's equations , vector field , magnetic field , physics , partial differential equation , geometry , mathematical optimization , quantum mechanics , thermodynamics
Summary 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.