Numerical analysis of Backward-Euler discretization for simplified magnetohydrodynamic flows (original) (raw)
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International Journal of Numerical Analysis and Modeling
We consider the finite element method for time dependent MHD ow at small magnetic Reynolds number. We make a second (and common) simplification in the model by assuming the time scales of the electrical and magnetic components are such that the electrical field responds instantaneously to changes in the uid motion. This report gives a comprehensive error analysis for both the semi-discrete and a fully-discrete approximation. Finally, the efiectiveness of the method is illustrated in several numeral experiments.
A finite element method for magnetohydrodynamics
Computer Methods in Applied Mechanics and Engineering, 2001
This paper presents a ®nite element method for the solution of 3D incompressible magnetohydrodynamic (MHD)¯ows. Two important issues are thoroughly addressed. First, appropriate formulations for the magnetic governing equations and the corresponding weak variational forms are discussed. The selected B; q formulation is conservative in the sense that the local divergencefree condition of the magnetic ®eld is accounted for in the variational sense. A Galerkin-least-squares variational formulation is used allowing equal-order approximations for all unknowns. In the second issue, a solution algorithm is developed for the solution of the coupled problem which is valid for both high and low magnetic Reynolds numbers. Several numerical benchmark tests are carried out to assess the stability and accuracy of the ®nite element method and to test the behavior of the solution algorithm. Ó
Applied Mathematics and Computation, 2007
This paper presents numerical simulations of incompressible fluid flows in the presence of a magnetic field at low magnetic Reynolds number. The equations governing the flow are the Navier–Stokes equations of fluid motion coupled with Maxwell’s equations of electromagnetics. The study of fluid flows under the influence of a magnetic field and with no free electric charges or electric fields is known as magnetohydrodynamics. The magnetohydrodynamics approximation is considered for the formulation of the non-dimensional problem and for the characterization of similarity parameters. A finite-difference technique is used to discretize the equations. In particular, an extension of the generalized Peaceman and Rachford alternating-direction implicit (ADI) scheme for simulating two-dimensional fluid flows is presented. The discretized conservation equations are solved in stream function–vorticity formulation. We compare the ADI and generalized ADI schemes, and show that the latter is more efficient in simulating low Reynolds number and magnetic Reynolds number problems. Numerical results demonstrating the applicability of this technique are also presented. The simulation of incompressible magnetohydrodynamic fluid flows is illustrated by numerical solution for two-dimensional cases.
Finite element method in applications of magnetohydrodynamics
The magnetically induced flow was examined numerically using a computational code based on the finite element method with the streamline-upwind/pressure-stabilized Petrov-Galerkin approach. The mathematical model considers an incompressible unsteady flow with a low frequency and low induction magnetic field. The validation of the magnetic force calculation was carried out on a cylindrical cavity, where the time-dependent electric potential and current density distribution can be derived analytically. The flow under the rotating magnetic field was simulated for the axisymmetric cylindrical and non-axisymmetric square cavity. The effect of the different geometries on the distribution of the time-averaged magnetic force and magnetically driven rotating flow was discussed.
Stability of partitioned methods for magnetohydrodynamics flows at small magnetic Reynolds number
Recent Advances in Scientific Computing and Applications, 2013
MHD flows are governed by the Navier-Stokes equations coupled with the Maxwell equations. Broadly, MHD flows in astrophysics occur at large magnetic Reynolds numbers while those in terrestrial applications, such as liquid metals, occur at small magnetic Reynolds numbers, the case considered herein. The physical processes of fluid flows and electricity and magnetism are quite different and numerical simulations of non-model problems can require different meshes, time steps and methods. We introduce implicit-explicit (IMEX) methods where the MHD equations can be evolved in time by calls to the NSE and Maxwell codes, each possibly optimized for the subproblem's respective physics.
Mugla Journal of Science and Technology, 2021
In this study, the solutions of Simplified Magnetohyrodynamics (SMHD) equations by finite element method are examined with nonlinear time relaxation term. The differential filter κ(|u-u ̅ |(u-u ̅ )) term is added to SMHD equations. Also SMHD Nonlinear Time Relaxation Model (SMHDNTRM) is introduced. The model is discretized by Backward-Euler (BE) method to obtain the finite element solutions. Moreover, the stability of the method is proved. The method is found unconditionally stable. The effectiveness of the method is exemplified by several cases with comparing different methods. FreeFem++ is used for all computations.
Computer Methods in Applied Mechanics and Engineering, 2010
We introduce and analyze a mixed finite element method for the numerical discretization of a stationary incompressible magnetohydrodynamics problem, in two and three dimensions. The velocity field is discretized using divergence-conforming Brezzi-Douglas-Marini (BDM) elements and the magnetic field is approximated by curl-conforming Nédélec elements. The H 1-continuity of the velocity field is enforced by a DG approach. A central feature of the method is that it produces exactly divergence-free velocity approximations, and captures the strongest magnetic singularities. We prove that the energy norm error is convergent in the mesh size in general Lipschitz polyhedra under minimal regularity assumptions, and derive nearly optimal a priori error estimates for the two-dimensional case. We present a comprehensive set of numerical experiments, which indicate optimal convergence of the proposed method for twodimensional as well as three-dimensional problems.
Stable discretization of magnetohydrodynamics in bounded domains
Communications in Mathematical Sciences, 2010
We study a semi-implicit time-difference scheme for magnetohydrodynamics of a viscous and resistive incompressible fluid in a bounded smooth domain with perfectly conducting boundary. In the scheme, velocity and magnetic fields are updated by solving simple Helmholtz equations. Pressure is treated explicitly in time, by solving Poisson equations corresponding to a recently developed formula for the Navier-Stokes pressure involving the commutator of Laplacian and Leray projection operators. We prove stability of the time-difference scheme, and deduce a local-time well-posedness theorem for MHD dynamics extended to ignore the divergence-free constraint on velocity and magnetic fields. These fields are divergence-free for all later time if they are initially so.
An extensive numerical benchmark of the various magnetohydrodynamic flows
International Journal of Heat and Fluid Flow, 2021
There is a continuous need for an updated series of numerical benchmarks dealing with various aspects of the magnetohydrodynamics (MHD) phenomena (i.e. interactions of the flow of an electrically conducting fluid and an externally imposed magnetic field). The focus of the present study is numerical magnetohydrodynamics (MHD) where we have performed an extensive series of simulations for generic configurations, including: (i) a laminar conjugate MHD flow in a duct with varied electrical conductivity of the walls, (ii) a back-step flow, (iii) a multiphase cavity flow, (iv) a rising bubble in liquid metal and (v) a turbulent conjugate MHD flow in a duct with varied electrical conductivity of surrounding walls. All considered benchmark situations are for the one-way coupled MHD approach, where the induced magnetic field is negligible. The governing equations describing the one-way coupled MHD phenomena are numerically implemented in the open-source code OpenFOAM. The novel elements of the numerical algorithm include fully-conservative forms of the discretized Lorentz force in the momentum equation and divergence-free current density, the conjugate MHD (coupling of the wall/fluid domains), the multi-phase MHD, and, finally, the MHD turbulence. The multi-phase phenomena are simulated with the Volume of Fluid (VOF) approach, whereas the MHD turbulence is simulated with the dynamic Large-Eddy Simulation (LES) method. For all considered benchmark cases, a very good agreement is obtained with available analytical solutions and other numerical results in the literature. The presented extensive numerical benchmarks are expected to be potentially useful for developers of the numerical codes used to simulate various types of the complex MHD phenomena.
A Monolithic Finite Element Formulation for Magnetohydrodynamics Involving a Compressible Fluid
Fluids, 2022
This work develops a new monolithic finite-element-based strategy for magnetohydrodynamics (MHD) involving a compressible fluid based on a continuous velocity–pressure formulation. The entire formulation is within a nodal finite element framework, and is directly in terms of physical variables. The exact linearization of the variational formulation ensures a quadratic rate of convergence in the vicinity of the solution. Both steady-state and transient formulations are presented for two- and three-dimensional flows. Several benchmark problems are presented, and comparisons are carried out against analytical solutions, experimental data, or against other numerical schemes for MHD. We show a good coarse-mesh accuracy and robustness of the proposed strategy, even at high Hartmann numbers.