An integro-differential formulation for magnetic induction in bounded domains: boundary element–finite volume method

Iskakov, Alexey B.; Descombes, Stéphane; Dormy, Emmanuel

doi:10.1016/j.jcp.2003.12.008

Cited by 47 publications

(55 citation statements)

References 23 publications

(44 reference statements)

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“…[17][18][19] In the following, we adopt a method proposed by Meir et al 20 for the quasi-stationary case, i.e., we compute the magnetic field using the Biot-Savart law…”

Section: Mathematical Modelmentioning

confidence: 99%

The influence of current collectors on Tayler instability and electro-vortex flows in liquid metal batteries

et al. 2015

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The Tayler instability is a kink-type flow instability which occurs when the electrical current through a conducting fluid exceeds a certain critical value. Originally studied in the astrophysical context, the instability was recently shown to be also a limiting factor for the upward scalability of liquid metal batteries. In this paper, we continue our efforts to simulate this instability for liquid metals within the framework of an integro-differential equation approach. The original solver is enhanced by multi-domain support with Dirichlet-Neumann partitioning for the static boundaries.Particular focus is laid on the detailed influence of the axial electrical boundary conditions on the characteristic features of the Tayler instability, and, secondly, on the occurrence of electro-vortex flows and their relevance for liquid metal batteries.

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“…[17][18][19] In the following, we adopt a method proposed by Meir et al 20 for the quasi-stationary case, i.e., we compute the magnetic field using the Biot-Savart law…”

Section: Mathematical Modelmentioning

confidence: 99%

The influence of current collectors on Tayler instability and electro-vortex flows in liquid metal batteries

et al. 2015

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“…Using this formulation [24], a stationary kinematic dynamo problem was solved in a cylindrical geometry. In [25,26], a finite volume method is used to discretize the solution in the interior, which is matched to that in the exterior vacuum via a boundary element method. An integral equation formulation was applied to the entire domain in [27], and [28] uses finite elements with a penalty method to apply boundary conditions.…”

Section: Towards An Mhd Solvermentioning

confidence: 99%

Poloidal–toroidal decomposition in a finite cylinder. I: Influence matrices for the magnetohydrodynamic equations

Boronski

Tuckerman

2007

Journal of Computational Physics

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The Navier-Stokes equations and magnetohydrodynamics equations are written in terms of poloidal and toroidal potentials in a finite cylinder. This formulation insures that the velocity and magnetic fields are divergence-free by construction, but leads to systems of partial differential equations of higher order, whose boundary conditions are coupled. The influence matrix technique is used to transform these systems into decoupled parabolic and elliptic problems. The magnetic field in the induction equation is matched to that in an exterior vacuum by means of the Dirichlet-to-Neumann mapping, thus eliminating the need to discretize the exterior. The influence matrix is scaled in order to attain an acceptable condition number. Motivation and Governing EquationsThe requirement that velocity and magnetic fields be solenoidal, i.e. divergence-free, represents one of the most challenging difficulties in hydrodynamics and in magnetohydrodynamics [1,2,3,4,5,6,7]. For the velocity field, this condition is the fundamental approximation used in incompressible fluid dynamics. For the magnetic field, this condition is the statement of the non-existence of magnetic monopoles.Two main approaches exist for imposing this requirement. The first is to use three field components and to project three-dimensional fields onto a divergence-free field. In an incompressible fluid, the pressure serves to counterbalance the nonlinear term which is the source of the divergence in the Navier-Stokes equations; the pressure also plays this role numerically. The divergence of the Navier-Stokes equations is taken, leading to a Poisson problem for the pressure. However, the boundary conditions on the

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“…Nevertheless we will mention several approaches which are used to compute MHD in wall-bounded geometry. A review of MHD solvers developed to compute fusion-plasma-related flows is given in [3], solvers aiming at a description of dynamo computations are, for example, given in [4,5,6] and computations investigating the magnetorotational instability in bounded domain were reported by Rüdiger and Shalybkov [7], Gissinger et al [8] and Willis and Barenghi [9]. An early numerical approach to study MHD in cylindrical geometry was proposed and validated by Shan et al [10] and more recently applied to spherical geometry by Mininni et al [11].…”

Section: Introductionmentioning

confidence: 99%

Simulation of confined magnetohydrodynamic flows with Dirichlet boundary conditions using a pseudo-spectral method with volume penalization

Moralès

Leroy

Bos

et al. 2014

Journal of Computational Physics

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A volume penalization approach to simulate magnetohydrodynamic (MHD) flows in confined domains is presented. Here the incompressible visco-resistive MHD equations are solved using parallel pseudo-spectral solvers in Cartesian geometries. The volume penalization technique is an immersed boundary method which is characterized by a high flexibility for the geometry of the considered flow. In the present case, it allows to use other than periodic boundary conditions in a Fourier pseudo-spectral approach. The numerical method is validated and its convergence is assessed for two-and three-dimensional hydrodynamic (HD) and MHD flows, by comparing the numerical results with results from literature and analytical solutions. The test cases considered are two-dimensional Taylor-Couette flow, the z-pinch configuration, three dimensional Orszag-Tang flow, ohmic-decay in a periodic cylinder, three-dimensional Taylor-Couette flow with and without axial magnetic field and three-dimensional Hartmann-instabilities in a cylinder with an imposed helical magnetic field.

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An integro-differential formulation for magnetic induction in bounded domains: boundary element–finite volume method

Cited by 47 publications

References 23 publications

The influence of current collectors on Tayler instability and electro-vortex flows in liquid metal batteries

The influence of current collectors on Tayler instability and electro-vortex flows in liquid metal batteries

Poloidal–toroidal decomposition in a finite cylinder. I: Influence matrices for the magnetohydrodynamic equations

Simulation of confined magnetohydrodynamic flows with Dirichlet boundary conditions using a pseudo-spectral method with volume penalization

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